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Mathematics · Textbook solutions

Vectors

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 235 questions

5.1 Vectors and their Types

40 q

Solved Examples

Worked · 16
  1. 5.1 SolvedEx.1
    In Fig. 5.30, five vectors a⃗,b⃗,c⃗,d⃗,e⃗\vec{a}, \vec{b}, \vec{c}, \vec{d}, \vec{e} are shown as directed line segments: a⃗\vec{a} and b⃗\vec{b} point to the right and are parallel to each other with b⃗\vec{b} drawn below a⃗\vec{a}; c⃗\vec{c} and d⃗\vec{d} point to the left and are parallel to each other with d⃗\vec{d} drawn below c⃗\vec{c}; e⃗\vec{e} is a short vector at the bottom pointing to the right, parallel to a⃗\vec{a} and b⃗\vec{b}. State the vectors which are: (i) equal in magnitude (ii) parallel (iii) in the same direction (iv) equal (v) negatives of one another.
  2. 5.1 SolvedEx.2
    In the diagram KL⃗=a⃗\vec{KL}=\vec{a}, LN⃗=b⃗\vec{LN}=\vec{b}, NM⃗=c⃗\vec{NM}=\vec{c} and KT⃗=d⃗\vec{KT}=\vec{d}. Find in terms of a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} and d⃗\vec{d}: (i) LT⃗\vec{LT} (ii) KM⃗\vec{KM} (iii) TN⃗\vec{TN} (iv) MT⃗\vec{MT}
  3. 5.1 SolvedEx.3
    Find the magnitude of the following vectors: (i) a⃗=i^−2j^+4k^\vec{a}=\hat{i}-2\hat{j}+4\hat{k} (ii) b⃗=4i^−3j^−7k^\vec{b}=4\hat{i}-3\hat{j}-7\hat{k} (iii) a vector with initial point (1,−3,4)(1,-3,4); terminal point (1,0,−1)(1,0,-1).
  4. 5.1 SolvedEx.4
    A(2, 3), B(-1, 5), C(-1, 1) and D(-7, 5) are four points in the Cartesian plane. (i) Find AB⃗\vec{AB} and CD⃗\vec{CD}. (ii) Check if CD⃗\vec{CD} is parallel to AB⃗\vec{AB}. (iii) E is the point (k,1)(k, 1) and AC⃗\vec{AC} is parallel to BE⃗\vec{BE}. Find kk.
  5. 5.1 SolvedEx.5
    Determine the values of cc that satisfy ∣cu⃗∣=3|c\vec{u}|=3, u⃗=i^+2j^+3k^\vec{u}=\hat{i}+2\hat{j}+3\hat{k}
  6. 5.1 SolvedEx.6
    Find a unit vector (i) in the direction of u⃗\vec{u} and (ii) in the direction opposite of u⃗\vec{u}, where u⃗=8i^+3j^−k^\vec{u}=8\hat{i}+3\hat{j}-\hat{k}.
  7. 5.1 SolvedEx.7
    Show that the vectors 2i^−3j^+4k^2\hat{i}-3\hat{j}+4\hat{k} and −4i^+6j^−8k^-4\hat{i}+6\hat{j}-8\hat{k} are parallel.
  8. 5.1 SolvedEx.8
    The non-zero vectors a⃗\vec{a} and b⃗\vec{b} are not collinear. Find the value of λ\lambda and μ\mu: (i) a⃗+3b⃗=2λa⃗−μb⃗\vec{a}+3\vec{b}=2\lambda\vec{a}-\mu\vec{b} (ii) (1+λ)a⃗+2λb⃗=μa⃗+4μb⃗(1+\lambda)\vec{a}+2\lambda\vec{b}=\mu\vec{a}+4\mu\vec{b} (iii) (3λ+5)a⃗+b⃗=2μa⃗+(λ−3)b⃗(3\lambda+5)\vec{a}+\vec{b}=2\mu\vec{a}+(\lambda-3)\vec{b}
  9. 5.1 SolvedEx.9
    Are the following sets of vectors linearly independent? (i) a⃗=i^−2j^+3k^\vec{a}=\hat{i}-2\hat{j}+3\hat{k}, b⃗=3i^−6j^+9k^\vec{b}=3\hat{i}-6\hat{j}+9\hat{k} (ii) a⃗=−2i^−4k^\vec{a}=-2\hat{i}-4\hat{k}, b⃗=i^−2j^−k^\vec{b}=\hat{i}-2\hat{j}-\hat{k}, c⃗=i^−4j^+3k^\vec{c}=\hat{i}-4\hat{j}+3\hat{k}. Interpret the results.
  10. 5.1 SolvedEx.10
    If a⃗=4i^+3k^\vec{a}=4\hat{i}+3\hat{k} and b⃗=−2i^+j^+5k^\vec{b}=-2\hat{i}+\hat{j}+5\hat{k}, find (i) ∣a⃗∣|\vec{a}|, (ii) a⃗+b⃗\vec{a}+\vec{b}, (iii) a⃗−b⃗\vec{a}-\vec{b}, (iv) 3b⃗3\vec{b}, (v) 2a⃗+5b⃗2\vec{a}+5\vec{b}
  11. 5.1 SolvedEx.11
    What is the distance from the point (2, 3, 4) to (i) the XY plane? (ii) the X-axis? (iii) origin (iv) point (-2, 7, 3).
  12. 5.1 SolvedEx.12
    Prove that the line segment joining the midpoints of two sides of a triangle is parallel to and half of the third side.
  13. 5.1 SolvedEx.13
    In quadrilateral ABCD, M and N are the mid-points of the diagonals AC and BD respectively. Prove that AB⃗+AD⃗+CB⃗+CD⃗=4MN⃗\vec{AB}+\vec{AD}+\vec{CB}+\vec{CD}=4\vec{MN}
  14. 5.1 SolvedEx.14
    Express −i^−3j^+4k^-\hat{i}-3\hat{j}+4\hat{k} as the linear combination of the vectors 2i^+j^−4k^2\hat{i}+\hat{j}-4\hat{k}, 2i^−j^+3k^2\hat{i}-\hat{j}+3\hat{k} and 3i^+j^−2k^3\hat{i}+\hat{j}-2\hat{k}.
  15. 5.1 SolvedEx.15
    Show that the three points A(1, -2, 3), B(2, 3, -4) and C(0, -7, 10) are collinear.
  16. 5.1 SolvedEx.16
    Show that the vectors 4i^+13j^−18k^4\hat{i}+13\hat{j}-18\hat{k}, i^−2j^+3k^\hat{i}-2\hat{j}+3\hat{k} and 2i^+3j^−4k^2\hat{i}+3\hat{j}-4\hat{k} are coplanar.

Exercise 5.1

Practice · 24
  1. Find the coordinates of the point which is located :
    5.1 Ex.Q9 a)
    Three units behind the YZ-plane, four units to the right of the XZ-plane and five units above the XY-plane.
  2. 5.1 Ex.Q9 b)
    In the YZ-plane, one unit to the right of the XZ-plane and six units above the XY-plane.
  3. 5.1 Ex.Q10
    Find the area of the triangle with vertices (1,1,0)(1, 1, 0), (1,0,1)(1, 0, 1) and (0,1,1)(0, 1, 1).
  4. 5.1 Ex.Q11
    If AB⃗=2i^−4j^+7k^\vec{AB} = 2\hat{i} - 4\hat{j} + 7\hat{k} and initial point A≡(1,5,0)A \equiv (1, 5, 0). Find the terminal point B.
  5. Show that the following points are collinear :
    5.1 Ex.Q12 i)
    A (3, 2, -4), B (9, 8, -10), C (-2, -3, 1).
  6. 5.1 Ex.Q12 ii)
    P (4, 5, 2), Q (3, 2, 4), R (5, 8, 0).
  7. 5.1 Ex.Q13
    If the vectors 2i^−qj^+3k^2\hat{i} - q\hat{j} + 3\hat{k} and 4i^−5j^+6k^4\hat{i} - 5\hat{j} + 6\hat{k} are collinear, then find the value of qq.
  8. 5.1 Ex.Q14
    Are the four points A(1, -1, 1), B(-1, 1, 1), C(1, 1, 1) and D(2, -3, 4) coplanar? Justify your answer.
  9. 5.1 Ex.Q15
    Express −i^−3j^+4k^-\hat{i} - 3\hat{j} + 4\hat{k} as linear combination of the vectors 2i^+j^−4k^2\hat{i} + \hat{j} - 4\hat{k}, 2i^−j^+3k^2\hat{i} - \hat{j} + 3\hat{k} and 3i^+j^−2k^3\hat{i} + \hat{j} - 2\hat{k}.
  10. 5.1 Ex.Q1
    The vector a⃗\vec{a} is directed due north and ∣a⃗∣=24|\vec{a}|=24. The vector b⃗\vec{b} is directed due west and ∣b⃗∣=7|\vec{b}|=7. Find ∣a⃗+b⃗∣|\vec{a}+\vec{b}|.
  11. In the triangle PQR, PQ⃗=2a⃗\vec{PQ}=2\vec{a} and QR⃗=2b⃗\vec{QR}=2\vec{b}. The mid-point of PR is M. Find the following vectors in terms of a⃗\vec{a} and b⃗\vec{b}.
    5.1 Ex.Q2 i)
    PR⃗\vec{PR}
  12. 5.1 Ex.Q2 ii)
    PM⃗\vec{PM}
  13. 5.1 Ex.Q2 iii)
    QM⃗\vec{QM}
  14. 5.1 Ex.Q3
    OABCDE is a regular hexagon. The points A and B have position vectors a⃗\vec{a} and b⃗\vec{b} respectively, referred to the origin O. Find, in terms of a⃗\vec{a} and b⃗\vec{b}, the position vectors of C, D and E.
  15. 5.1 Ex.Q4
    If ABCDEF is a regular hexagon, show that AB⃗+AC⃗+AD⃗+AE⃗+AF⃗=6AO⃗\vec{AB}+\vec{AC}+\vec{AD}+\vec{AE}+\vec{AF}=6\vec{AO}, where O is the centre of the hexagon.
  16. 5.1 Ex.Q5
    Check whether the vectors 2i^+2j^+3k^2\hat{i}+2\hat{j}+3\hat{k}, −3i^+3j^+2k^-3\hat{i}+3\hat{j}+2\hat{k} and 3i^+4k^3\hat{i}+4\hat{k} form a triangle or not.
  17. 5.1 Ex.Q6
    In the figure 5.34, express c⃗\vec{c} and d⃗\vec{d} in terms of a⃗\vec{a} and b⃗\vec{b}.
  18. 5.1 Ex.Q7
    Find a vector in the direction of a⃗=i^−2j^\vec{a}=\hat{i}-2\hat{j} that has magnitude 7 units.
  19. Find the distance from (4, -2, 6) to each of the following:
    5.1 Ex.Q8 a)
    The XY-plane
  20. 5.1 Ex.Q8 b)
    The YZ-plane
  21. 5.1 Ex.Q8 c)
    The XZ-plane
  22. 5.1 Ex.Q8 d)
    The X-axis
  23. 5.1 Ex.Q8 e)
    The Y-axis
  24. 5.1 Ex.Q8 f)
    The Z-axis

5.2 Section Formula

27 q

Solved Examples

Worked · 13
  1. Find the co-ordinates of the point which divides the line segment joining the points A(2, –6, 8) and B(–1, 3, –4).
    5.2 SolvedEx.1 i)
    (i) Internally in the ratio 1 : 3.
  2. 5.2 SolvedEx.1 ii)
    (ii) Externally in the ratio 1 : 3.
  3. 5.2 SolvedEx.2
    If the three points A(3, 2, p), B(q, 8, –10), C(–2, –3, 1) are collinear then find (i) the ratio in which the point C divides the line segment AB, (ii) the values of p and q.
  4. 5.2 SolvedEx.3
    If A(5, 1, p), B(1, q, p) and C(1, –2, 3) are vertices of a triangle and G(r,−43,13)G\left(r, -\dfrac{4}{3}, \dfrac{1}{3}\right) is its centroid, then find the values of p, q and r.
  5. 5.2 SolvedEx.4
    If a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c} are the position vectors of the points A, B, C respectively and 5a⃗−3b⃗−2c⃗=0⃗5\vec{a} - 3\vec{b} - 2\vec{c} = \vec{0}, then find the ratio in which the point C divides the line segment BA.
  6. 5.2 SolvedEx.5
    Prove that the medians of a triangle are concurrent.
  7. 5.2 SolvedEx.6
    Prove that the angle bisectors of a triangle are concurrent.
  8. 5.2 SolvedEx.7
    Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4).
  9. 5.2 SolvedEx.8
    If 4i^+7j^+8k^4\hat{i}+7\hat{j}+8\hat{k}, 2i^+3j^+4k^2\hat{i}+3\hat{j}+4\hat{k} and 2i^+5j^+7k^2\hat{i}+5\hat{j}+7\hat{k} are the position vectors of the vertices A, B and C respectively of triangle ABC, find the position vector of the point in which the bisector of ∠A\angle A meets BC.
  10. 5.2 SolvedEx.9
    If G(a, 2, –1) is the centroid of the triangle with vertices P(1, 3, 2), Q(3, b, –4) and R(5, 1, c), then find the values of a, b and c.
  11. 5.2 SolvedEx.10
    Find the centroid of the tetrahedron with vertices A(3, –5, 7), B(5, 4, 2), C(7, –7, –3), D(1, 0, 2).
  12. 5.2 SolvedEx.11
    Find the ratio in which point P divides AB and CD where A(2, –3, 4), B(0, 5, 2), C(–1, 5, 3) and D(2, –1, 3). Also, find its coordinates.
  13. 5.2 SolvedEx.12
    In a triangle ABC, D and E are points on BC and AC respectively, such that BD = 2DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PF using vector methods.

Exercise 5.2

Practice · 14
  1. Find the position vector of point R which divides the line joining the points P and Q whose position vectors are 2i^−j^+3k^2\hat{i}-\hat{j}+3\hat{k} and −5i^+2j^−5k^-5\hat{i}+2\hat{j}-5\hat{k} in the ratio 3 : 2.
    5.2 Ex.Q1 i)
    (i) Internally.
  2. 5.2 Ex.Q1 ii)
    (ii) Externally.
  3. 5.2 Ex.Q2
    Find the position vector of mid-point M joining the points L(7, –6, 12) and N(5, 4, –2).
  4. If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find
    5.2 Ex.Q3 i)
    (i) The ratio in which the point C divides the line segment AB.
  5. 5.2 Ex.Q3 ii)
    (ii) The values of p and q.
  6. 5.2 Ex.Q4
    The position vectors of points A and B are 6a⃗+2b⃗6\vec{a}+2\vec{b} and a⃗−3b⃗\vec{a}-3\vec{b}. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is 3a⃗−b⃗3\vec{a}-\vec{b}.
  7. 5.2 Ex.Q5
    Prove that the line segments joining mid-points of adjacent sides of a quadrilateral form a parallelogram.
  8. 5.2 Ex.Q6
    D and E divide sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE.
  9. 5.2 Ex.Q7
    Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
  10. 5.2 Ex.Q8
    Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half the sum of the parallel sides.
  11. 5.2 Ex.Q9
    If two of the vertices of the triangle are A(3, 1, 4) and B(–4, 5, –3) and the centroid of the triangle is G(–1, 2, 1), then find the coordinates of the third vertex C of the triangle.
  12. 5.2 Ex.Q10
    In △OAB\triangle OAB, E is the mid-point of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.
  13. 5.2 Ex.Q11
    If the centroid of a tetrahedron OABC is (1, 2, –1) where A = (a, 2, 3), B = (1, b, 2), C = (2, 1, c) respectively, find the distance of P(a, b, c) from the origin.
  14. 5.2 Ex.Q12
    Find the centroid of tetrahedron with vertices K(5, –7, 0), L(1, 5, 3), M(4, –6, 3), N(6, –4, 2)?

5.3 Dot Product of Vectors

34 q

Solved Examples

Worked · 16
  1. 5.3 SolvedEx.1
    Find a⃗⋅b⃗\vec{a} \cdot \vec{b} if ∣a⃗∣=3,∣b⃗∣=6|\vec{a}| = 3, |\vec{b}| = \sqrt{6}, the angle between a⃗\vec{a} and b⃗\vec{b} is 45∘45^\circ.
  2. If a⃗=3i^+4j^−5k^\vec{a} = 3\hat{i} + 4\hat{j} - 5\hat{k} and b⃗=3i^−4j^−5k^\vec{b} = 3\hat{i} - 4\hat{j} - 5\hat{k}
    5.3 SolvedEx.2 i)
    Find a⃗⋅b⃗\vec{a} \cdot \vec{b}.
  3. 5.3 SolvedEx.2 ii)
    Find the angle between a⃗\vec{a} and b⃗\vec{b}.
  4. 5.3 SolvedEx.2 iii)
    Find the scalar projection of a⃗\vec{a} in the direction of b⃗\vec{b}.
  5. 5.3 SolvedEx.2 iv)
    Find the vector projection of b⃗\vec{b} along a⃗\vec{a}.
  6. Find the value of aa for which the vectors 3i^+2j^+9k^3\hat{i} + 2\hat{j} + 9\hat{k} and i^+aj^+3k^\hat{i} + a\hat{j} + 3\hat{k} are
    5.3 SolvedEx.3 i)
    perpendicular.
  7. 5.3 SolvedEx.3 ii)
    parallel.
  8. 5.3 SolvedEx.4
    If a⃗=i^+2j^−3k^\vec{a} = \hat{i}+2\hat{j}-3\hat{k} and b⃗=3i^−j^+2k^\vec{b} = 3\hat{i}-\hat{j}+2\hat{k} find the angle between the vectors 2a⃗+b⃗2\vec{a}+\vec{b} and a⃗+2b⃗\vec{a}+2\vec{b}.
  9. 5.3 SolvedEx.5
    If a line makes angle 90∘,60∘90^\circ, 60^\circ and 30∘30^\circ with the positive direction of X, Y and Z axes respectively, find its direction cosines.
  10. 5.3 SolvedEx.6
    Find the vector projection of PQ⃗\vec{PQ} on AB⃗\vec{AB} where P, Q, A, B are the points (−2,1,3)(-2, 1, 3), (3,2,5)(3, 2, 5), (4,−3,5)(4, -3, 5) and (7,−5,−1)(7, -5, -1) respectively.
  11. 5.3 SolvedEx.7
    Find the values of λ\lambda for which the angle between the vectors a⃗=2λ2i^+4λj^+k^\vec{a} = 2\lambda^2\hat{i}+4\lambda\hat{j}+\hat{k} and b⃗=7i^−2j^+λk^\vec{b} = 7\hat{i}-2\hat{j}+\lambda\hat{k} is obtuse.
  12. 5.3 SolvedEx.8
    Find the direction cosines of the vector 2i^+2j^−k^2\hat{i}+2\hat{j}-\hat{k}.
  13. 5.3 SolvedEx.9
    Find the position vector of a point P such that AB⃗\vec{AB} is inclined to the X axis at 45∘45^\circ and to the Y axis at 60∘60^\circ, and OP=12OP = 12 units.
  14. 5.3 SolvedEx.10
    A line makes angles of measure 45∘45^\circ and 60∘60^\circ with the positive direction of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.
  15. 5.3 SolvedEx.11
    A line passes through the points (6,−7,−1)(6, -7, -1) and (2,−3,1)(2, -3, 1). Find the direction ratios and the direction cosines of the line so that the angle α\alpha is acute.
  16. 5.3 SolvedEx.12
    Prove that the altitudes of a triangle are concurrent.

Exercise 5.3

Practice · 18
  1. 5.3 Ex.Q12
    If a line has the direction ratios, 4, -12, 18 then find its direction cosines.
  2. 5.3 Ex.Q13
    The direction ratios of AB⃗\vec{AB} are -2, 2, 1. If A = (4, 1, 5) and l(AB)=6l(AB) = 6 units, find B.
  3. 5.3 Ex.Q14
    Find the angle between the lines whose direction cosines l,m,nl, m, n satisfy the equations 5l+m+3n=05l + m + 3n = 0 and 5mn−2nl+6lm=05mn - 2nl + 6lm = 0.
  4. 5.3 Ex.Q1
    Find two unit vectors each of which is perpendicular to both u⃗\vec{u} and v⃗\vec{v}, where u⃗=2i^+j^−2k^,v⃗=i^+2j^−2k^\vec{u} = 2\hat{i}+\hat{j}-2\hat{k}, \vec{v} = \hat{i}+2\hat{j}-2\hat{k}.
  5. 5.3 Ex.Q2
    If a⃗\vec{a} and b⃗\vec{b} are two vectors perpendicular to each other, prove that (a⃗+b⃗)2=(a⃗−b⃗)2(\vec{a}+\vec{b})^2 = (\vec{a}-\vec{b})^2.
  6. 5.3 Ex.Q3
    Find the values of cc so that for all real xx the vectors xci^−6j^+3k^xc\hat{i}-6\hat{j}+3\hat{k} and xi^+2j^+2cxk^x\hat{i}+2\hat{j}+2cx\hat{k} make an obtuse angle.
  7. 5.3 Ex.Q4
    Show that the sum of the length of projections of pi^+qj^+rk^p\hat{i}+q\hat{j}+r\hat{k} on the coordinate axes, where p=2,q=3p=2, q=3 and r=4r=4, is 9.
  8. 5.3 Ex.Q5
    Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.
  9. Determine whether a⃗\vec{a} and b⃗\vec{b} are orthogonal, parallel or neither.
    5.3 Ex.Q6 i)
    a⃗=−9i^+6j^+15k^,b⃗=6i^−4j^−10k^\vec{a} = -9\hat{i}+6\hat{j}+15\hat{k}, \vec{b} = 6\hat{i}-4\hat{j}-10\hat{k}
  10. 5.3 Ex.Q6 ii)
    a⃗=2i^+3j^−k^,b⃗=5i^−2j^+4k^\vec{a} = 2\hat{i}+3\hat{j}-\hat{k}, \vec{b} = 5\hat{i}-2\hat{j}+4\hat{k}
  11. 5.3 Ex.Q6 iii)
    a⃗=−35i^+12j^+13k^,b⃗=5i^+4j^+3k^\vec{a} = -\frac{3}{5}\hat{i}+\frac{1}{2}\hat{j}+\frac{1}{3}\hat{k}, \vec{b} = 5\hat{i}+4\hat{j}+3\hat{k}
  12. 5.3 Ex.Q6 iv)
    a⃗=4i^−j^+6k^,b⃗=5i^−2j^+4k^\vec{a} = 4\hat{i}-\hat{j}+6\hat{k}, \vec{b} = 5\hat{i}-2\hat{j}+4\hat{k}
  13. 5.3 Ex.Q7
    Find the angle P of the triangle whose vertices are P(0,−1,−2)(0, -1, -2), Q(3,1,4)(3, 1, 4) and R(5,7,1)(5, 7, 1).
  14. If p^,q^\hat{p}, \hat{q} and r^\hat{r} are unit vectors such that p^+r^=q^\hat{p}+\hat{r}=\hat{q}, find (see fig. 5.50).
    5.3 Ex.Q8 i)
    p^⋅q^\hat{p}\cdot\hat{q}
  15. 5.3 Ex.Q8 ii)
    q^⋅r^\hat{q}\cdot\hat{r}
  16. 5.3 Ex.Q9
    Prove by vector method that the angle subtended on a semicircle is a right angle.
  17. 5.3 Ex.Q10
    If a vector has direction angles 45∘45^\circ and 60∘60^\circ, find the third direction angle.
  18. 5.3 Ex.Q11
    If a line makes angles 90∘,135∘,45∘90^\circ, 135^\circ, 45^\circ with the X, Y and Z axes respectively, then find its direction cosines.

5.4 Vector Product of Vectors

26 q

Solved Examples

Worked · 16
  1. Find the cross product a⃗×b⃗\vec{a} \times \vec{b} and verify that it is orthogonal (perpendicular) to both a⃗\vec{a} and b⃗\vec{b}.
    5.4 SolvedEx.1 i)
    a⃗=i^+j^−k^\vec{a} = \hat{i} + \hat{j} - \hat{k}, b⃗=2i^+4j^+6k^\vec{b} = 2\hat{i} + 4\hat{j} + 6\hat{k}
  2. 5.4 SolvedEx.1 ii)
    a⃗=i^+3j^−2k^\vec{a} = \hat{i} + 3\hat{j} - 2\hat{k}, b⃗=−i^+5k^\vec{b} = -\hat{i} + 5\hat{k}
  3. 5.4 SolvedEx.2
    Find all vectors of magnitude 10310\sqrt{3} that are perpendicular to the plane of i^+2j^+k^\hat{i}+2\hat{j}+\hat{k} and −i^+3j^+4k^-\hat{i}+3\hat{j}+4\hat{k}.
  4. 5.4 SolvedEx.3
    If u⃗+v⃗+w⃗=0⃗\vec{u}+\vec{v}+\vec{w}=\vec{0}, show that u⃗×v⃗=v⃗×w⃗=w⃗×u⃗\vec{u}\times\vec{v}=\vec{v}\times\vec{w}=\vec{w}\times\vec{u}.
  5. 5.4 SolvedEx.4
    If a⃗=3i^−j^+2k^\vec{a}=3\hat{i}-\hat{j}+2\hat{k}, b⃗=2i^+j^−k^\vec{b}=2\hat{i}+\hat{j}-\hat{k} and c⃗=i^−2j^+2k^\vec{c}=\hat{i}-2\hat{j}+2\hat{k}, find (a⃗×b⃗)×c⃗(\vec{a}\times\vec{b})\times\vec{c} and a⃗×(b⃗×c⃗)\vec{a}\times(\vec{b}\times\vec{c}) and hence show that (a⃗×b⃗)×c⃗≠a⃗×(b⃗×c⃗)(\vec{a}\times\vec{b})\times\vec{c}\neq\vec{a}\times(\vec{b}\times\vec{c}).
  6. 5.4 SolvedEx.5
    Find the area of the triangle with vertices (1,2,0), (1,0,2), and (0,3,1).
  7. 5.4 SolvedEx.6
    Find the area of the parallelogram with vertices K(1, 2, 3), L(1, 3, 6), M(3, 8, 6) and N(3, 7, 3).
  8. Find ∣u⃗×v⃗∣|\vec{u}\times\vec{v}| if
    5.4 SolvedEx.7 i)
    ∣u⃗∣=4|\vec{u}|=4, ∣v⃗∣=5|\vec{v}|=5, with the angle between u⃗\vec{u} and v⃗\vec{v} equal to 45°45° (as shown in Fig 5.54(i)).
  9. 5.4 SolvedEx.7 ii)
    ∣u⃗∣=12|\vec{u}|=12, ∣v⃗∣=16|\vec{v}|=16 (as shown in Fig 5.54(ii)).
  10. 5.4 SolvedEx.8
    Show that (a⃗−b⃗)×(a⃗+b⃗)=2(a⃗×b⃗)(\vec{a}-\vec{b})\times(\vec{a}+\vec{b}) = 2(\vec{a}\times\vec{b}).
  11. 5.4 SolvedEx.9
    Show that the three points with position vectors 3i^−2j^+4k^3\hat{i}-2\hat{j}+4\hat{k}, i^+j^+k^\hat{i}+\hat{j}+\hat{k} and −i^+4j^−2k^-\hat{i}+4\hat{j}-2\hat{k} respectively are collinear.
  12. 5.4 SolvedEx.10
    Find a unit vector perpendicular to PQ‾\overline{PQ} and PR‾\overline{PR} where P≡(2,2,0)P\equiv(2,2,0), Q≡(0,3,5)Q\equiv(0,3,5) and R≡(5,0,3)R\equiv(5,0,3). Also find the sine of angle between PQ‾\overline{PQ} and PR‾\overline{PR}.
  13. 5.4 SolvedEx.11
    If ∣a⃗∣=5|\vec{a}|=5, ∣b⃗∣=13|\vec{b}|=13 and ∣a⃗×b⃗∣=25|\vec{a}\times\vec{b}|=25, find a⃗⋅b⃗\vec{a}\cdot\vec{b}.
  14. 5.4 SolvedEx.12
    Direction ratios of two lines satisfy the relation 2a−b+2c=02a-b+2c=0 and ab+bc+ca=0ab+bc+ca=0. Show that the lines are perpendicular.
  15. 5.4 SolvedEx.13
    Find the direction cosines of the line which is perpendicular to the lines with direction ratios −1,2,2-1, 2, 2 and 0,2,10, 2, 1.
  16. 5.4 SolvedEx.14
    If M is the foot of the perpendicular drawn from A(4, 3, 2) on the line joining the points B(2, 4, 1) and C (4, 5, 3), find the coordinates of M.

Exercise 5.4

Practice · 10
  1. 5.4 Ex.Q1
    If a⃗=2i^+3j^−k^\vec{a}=2\hat{i}+3\hat{j}-\hat{k}, b⃗=i^−4j^+2k^\vec{b}=\hat{i}-4\hat{j}+2\hat{k} find (a⃗+b⃗)×(a⃗−b⃗)(\vec{a}+\vec{b})\times(\vec{a}-\vec{b}).
  2. 5.4 Ex.Q2
    Find unit vectors perpendicular to the vectors j^+2k^\hat{j}+2\hat{k} and i^+j^\hat{i}+\hat{j}.
  3. 5.4 Ex.Q3
    If a⃗⋅b⃗=3\vec{a}\cdot\vec{b} = \sqrt{3} and a⃗×b⃗=2i^+j^+2k^\vec{a}\times\vec{b} = 2\hat{i}+\hat{j}+2\hat{k}, find the angle between a⃗\vec{a} and b⃗\vec{b}.
  4. 5.4 Ex.Q4
    If a⃗=2i^+j^−3k^\vec{a}=2\hat{i}+\hat{j}-3\hat{k} and b⃗=i^−2j^+k^\vec{b}=\hat{i}-2\hat{j}+\hat{k}, find vectors of magnitude 5 perpendicular to both a⃗\vec{a} and b⃗\vec{b}.
  5. Find
    5.4 Ex.Q5 i)
    u⃗⋅v⃗\vec{u}\cdot\vec{v} if ∣u⃗∣=2|\vec{u}|=2, ∣v⃗∣=5|\vec{v}|=5, ∣u⃗×v⃗∣=8|\vec{u}\times\vec{v}|=8
  6. 5.4 Ex.Q5 ii)
    ∣u⃗×v⃗∣|\vec{u}\times\vec{v}| if ∣u⃗∣=10|\vec{u}|=10, ∣v⃗∣=2|\vec{v}|=2, u⃗⋅v⃗=12\vec{u}\cdot\vec{v}=12
  7. 5.4 Ex.Q6
    Prove that 2(a⃗−b⃗)×2(a⃗+b⃗)=8(a⃗×b⃗)2(\vec{a}-\vec{b})\times 2(\vec{a}+\vec{b}) = 8(\vec{a}\times\vec{b}).
  8. 5.4 Ex.Q7
    If a⃗=i^−2j^+3k^\vec{a}=\hat{i}-2\hat{j}+3\hat{k}, b⃗=4i^−3j^+k^\vec{b}=4\hat{i}-3\hat{j}+\hat{k} and c⃗=i^−j^+2k^\vec{c}=\hat{i}-\hat{j}+2\hat{k} verify that a⃗×(b⃗+c⃗)=a⃗×b⃗+a⃗×c⃗\vec{a}\times(\vec{b}+\vec{c}) = \vec{a}\times\vec{b}+\vec{a}\times\vec{c}.
  9. 5.4 Ex.Q8
    Find the area of the parallelogram whose adjacent sides are the vectors a⃗=2i^−2j^+k^\vec{a}=2\hat{i}-2\hat{j}+\hat{k} and b⃗=i^−3j^−3k^\vec{b}=\hat{i}-3\hat{j}-3\hat{k}.
  10. 5.4 Ex.Q9
    Show that vector area of a quadrilateral ABCDABCD is 12(AC‾×BD‾)\dfrac{1}{2}(\overline{AC}\times\overline{BD}), where AC and BD are its diagonals.

5.5 Scalar and Vector Triple Product

21 q

Solved Examples

Worked · 8
  1. Find the volume of the parallelepiped determined by the vectors a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c}.
    5.5 SolvedEx.1 i)
    a⃗=i^+2j^+3k^,b⃗=−i^+j^+2k^,c⃗=2i^+j^+4k^\vec{a}=\hat{i}+2\hat{j}+3\hat{k}, \vec{b}=-\hat{i}+\hat{j}+2\hat{k}, \vec{c}=2\hat{i}+\hat{j}+4\hat{k}
  2. 5.5 SolvedEx.1 ii)
    a⃗=i^+j^,b⃗=j^+k^,c⃗=i^+j^+k^\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{j}+\hat{k}, \vec{c}=\hat{i}+\hat{j}+\hat{k}
  3. 5.5 SolvedEx.2
    Find the scalar triple product [u⃗ v⃗ w⃗][\vec{u}\ \vec{v}\ \vec{w}] and verify that the vectors u⃗=i^+5j^−2k^,v⃗=3i^−j^\vec{u}=\hat{i}+5\hat{j}-2\hat{k}, \vec{v}=3\hat{i}-\hat{j} and w⃗=5i^+9j^−4k^\vec{w}=5\hat{i}+9\hat{j}-4\hat{k} are coplanar.
  4. 5.5 SolvedEx.3
    Find the vector which is orthogonal to the vector 3i^+2j^+6k^3\hat{i}+2\hat{j}+6\hat{k} and is coplanar with the vectors 2i^+j^+k^2\hat{i}+\hat{j}+\hat{k} and i^−j^+k^\hat{i}-\hat{j}+\hat{k}.
  5. 5.5 SolvedEx.4
    If a⃗=i^+j^+k^,a⃗⋅b⃗=1\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{a}\cdot\vec{b}=1 and a⃗×b⃗=j^−k^\vec{a}\times\vec{b}=\hat{j}-\hat{k} then prove that b⃗=i^\vec{b}=\hat{i}.
  6. 5.5 SolvedEx.5
    Prove that: a⃗×(b⃗×c⃗)+b⃗×(c⃗×a⃗)+c⃗×(a⃗×b⃗)=0⃗\vec{a}\times(\vec{b}\times\vec{c}) + \vec{b}\times(\vec{c}\times\vec{a}) + \vec{c}\times(\vec{a}\times\vec{b}) = \vec{0}
  7. 5.5 SolvedEx.6
    Show that the points A(2,−1,0),B(−3,0,4),C(−1,−1,4)A(2,-1,0), B(-3,0,4), C(-1,-1,4) and D(0,−5,2)D(0,-5,2) are non coplanar.
  8. 5.5 SolvedEx.7
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are non coplanar vectors, then show that the four points 2a⃗+b⃗,a⃗+2b⃗+c⃗,4a⃗−2b⃗−c⃗2\vec{a}+\vec{b}, \vec{a}+2\vec{b}+\vec{c}, 4\vec{a}-2\vec{b}-\vec{c} and 3a⃗+4b⃗−5c⃗3\vec{a}+4\vec{b}-5\vec{c} are coplanar.

Exercise 5.5

Practice · 13
  1. 5.5 Ex.Q1
    Find a⃗⋅(b⃗×c⃗)\vec{a}\cdot(\vec{b}\times\vec{c}), if a⃗=3i^−j^+4k^,b⃗=2i^+3j^−k^\vec{a}=3\hat{i}-\hat{j}+4\hat{k}, \vec{b}=2\hat{i}+3\hat{j}-\hat{k} and c⃗=−5i^+2j^+3k^\vec{c}=-5\hat{i}+2\hat{j}+3\hat{k}
  2. 5.5 Ex.Q2
    If the vectors 3i^+5k^,4i^+2j^−3k^3\hat{i}+5\hat{k}, 4\hat{i}+2\hat{j}-3\hat{k} and 3i^+j^+4k^3\hat{i}+\hat{j}+4\hat{k} are co-terminus edges of the parallelepiped, then find the volume of the parallelepiped.
  3. 5.5 Ex.Q3
    If the vectors −3i^+4j^−2k^,i^+2k^-3\hat{i}+4\hat{j}-2\hat{k}, \hat{i}+2\hat{k} and i^−pj^\hat{i}-p\hat{j} are coplanar, then find the value of pp.
  4. Prove that:
    5.5 Ex.Q4 i)
    [a⃗  b⃗+c⃗  a⃗+b⃗+c⃗]=0[\vec{a}\ \ \vec{b}+\vec{c}\ \ \vec{a}+\vec{b}+\vec{c}] = 0
  5. 5.5 Ex.Q4 ii)
    [a⃗+2b⃗−c⃗  a⃗−b⃗  a⃗−b⃗−c⃗]=3[a⃗  b⃗  c⃗][\vec{a}+2\vec{b}-\vec{c}\ \ \vec{a}-\vec{b}\ \ \vec{a}-\vec{b}-\vec{c}] = 3[\vec{a}\ \ \vec{b}\ \ \vec{c}]
  6. 5.5 Ex.Q5
    If c⃗=3a⃗−2b⃗\vec{c}=3\vec{a}-2\vec{b} and [a⃗  b⃗+c⃗  a⃗+b⃗+c⃗]=0[\vec{a}\ \ \vec{b}+\vec{c}\ \ \vec{a}+\vec{b}+\vec{c}] = 0 then prove that [a⃗ b⃗ c⃗]=0[\vec{a}\ \vec{b}\ \vec{c}] = 0
  7. If u⃗=i^−2j^+k^,v⃗=3i^+k^\vec{u}=\hat{i}-2\hat{j}+\hat{k}, \vec{v}=3\hat{i}+\hat{k} and w⃗=j^−k^\vec{w}=\hat{j}-\hat{k} are given vectors, then find:
    5.5 Ex.Q6 i)
    [u⃗+w⃗]⋅[(u⃗×v⃗)×(v⃗×w⃗)][\vec{u}+\vec{w}]\cdot[(\vec{u}\times\vec{v})\times(\vec{v}\times\vec{w})]
  8. 5.5 Ex.Q6 ii)
    [u⃗×v⃗  u⃗×w⃗  v⃗×w⃗][\vec{u}\times\vec{v}\ \ \vec{u}\times\vec{w}\ \ \vec{v}\times\vec{w}]
  9. 5.5 Ex.Q7
    Find the volume of a tetrahedron whose vertices are A(−1,2,3),B(3,−2,1),C(2,1,3)A(-1,2,3), B(3,-2,1), C(2,1,3) and D(−1,−2,4)D(-1,-2,4).
  10. 5.5 Ex.Q8
    If a⃗=i^+2j^+3k^,b⃗=3i^+2j^+k^\vec{a}=\hat{i}+2\hat{j}+3\hat{k}, \vec{b}=3\hat{i}+2\hat{j}+\hat{k} and c⃗=2i^+j^+3k^\vec{c}=2\hat{i}+\hat{j}+3\hat{k} then verify that a⃗×(b⃗×c⃗)=(a⃗⋅c⃗)b⃗−(a⃗⋅b⃗)c⃗\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\cdot\vec{c})\vec{b} - (\vec{a}\cdot\vec{b})\vec{c}
  11. If a⃗=i^−2j^,b⃗=i^+2j^\vec{a}=\hat{i}-2\hat{j}, \vec{b}=\hat{i}+2\hat{j} and c⃗=2i^+j^−2k^\vec{c}=2\hat{i}+\hat{j}-2\hat{k} then find the following. Are the results the same? Justify.
    5.5 Ex.Q9 i)
    a⃗×(b⃗×c⃗)\vec{a}\times(\vec{b}\times\vec{c})
  12. 5.5 Ex.Q9 ii)
    (a⃗×b⃗)×c⃗(\vec{a}\times\vec{b})\times\vec{c}
  13. 5.5 Ex.Q10
    Show that a⃗×(b⃗×c⃗)+b⃗×(c⃗×a⃗)+c⃗×(a⃗×b⃗)=0⃗\vec{a}\times(\vec{b}\times\vec{c}) + \vec{b}\times(\vec{c}\times\vec{a}) + \vec{c}\times(\vec{a}\times\vec{b}) = \vec{0}

Miscellaneous Exercise 5

87 q

Select the correct answer

Practice · 20
  1. Misc I Q.1
    If ∣a⃗∣=2|\vec{a}| = 2, ∣b⃗∣=3|\vec{b}| = 3, ∣c⃗∣=4|\vec{c}| = 4 then [ a⃗+b⃗   b⃗+c⃗   c⃗−a⃗ ]\left[\ \vec{a}+\vec{b}\ \ \ \vec{b}+\vec{c}\ \ \ \vec{c}-\vec{a}\ \right] is equal to

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  2. Misc I Q.2
    If ∣a⃗∣=3|\vec{a}| = 3, ∣b⃗∣=4|\vec{b}| = 4, then the value of λ\lambda for which a⃗+λb⃗\vec{a}+\lambda\vec{b} is perpendicular to a⃗−λb⃗\vec{a}-\lambda\vec{b}, is

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  3. Misc I Q.3
    If sum of two unit vectors is itself a unit vector, then the magnitude of their difference is

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  4. Misc I Q.4
    If ∣a⃗∣=3|\vec{a}| = 3, ∣b⃗∣=5|\vec{b}| = 5, ∣c⃗∣=7|\vec{c}| = 7 and a⃗+b⃗+c⃗=0⃗\vec{a}+\vec{b}+\vec{c}=\vec{0}, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  5. Misc I Q.5
    The volume of tetrahedron whose vertices are (1,−6,10)(1,-6,10), (−1,−3,7)(-1,-3,7), (5,−1,λ)(5,-1,\lambda) and (7,−4,7)(7,-4,7) is 11 cu. units then the value of λ\lambda is

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  6. Misc I Q.6
    If α,β,γ\alpha, \beta, \gamma are direction angles of a line and α=60∘\alpha = 60^\circ, β=45∘\beta = 45^\circ, the γ=\gamma =

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  7. Misc I Q.7
    The distance of the point (3,4,5)(3, 4, 5) from Y-axis is

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  8. Misc I Q.8
    The line joining the points (−2,1,−8)(-2, 1, -8) and (a,b,c)(a, b, c) is parallel to the line whose direction ratios are 6, 2, 3. The value of a, b, c are

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  9. Misc I Q.9
    If cos⁡α,cos⁡β,cos⁡γ\cos\alpha, \cos\beta, \cos\gamma are the direction cosines of a line then the value of sin⁡2α+sin⁡2β+sin⁡2γ\sin^2\alpha + \sin^2\beta + \sin^2\gamma is

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  10. Misc I Q.10
    If l,m,nl, m, n are direction cosines of a line then li^+mj^+nk^l\hat{i}+m\hat{j}+n\hat{k} is

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  11. Misc I Q.11
    If ∣a⃗∣=3|\vec{a}| = 3 and −1≤k≤2-1 \le k \le 2, then ∣ka⃗∣|k\vec{a}| lies in the interval

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  12. Misc I Q.12
    Let α,β,γ\alpha, \beta, \gamma be distinct real numbers. The points with position vectors αi^+βj^+γk^\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}, βi^+γj^+αk^\beta\hat{i}+\gamma\hat{j}+\alpha\hat{k}, γi^+αj^+βk^\gamma\hat{i}+\alpha\hat{j}+\beta\hat{k}

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  13. Misc I Q.13
    Let p⃗\vec{p} and q⃗\vec{q} be the position vectors of P and Q respectively, with respect to O and ∣p⃗∣=p|\vec{p}| = p, ∣q⃗∣=q|\vec{q}| = q. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular then.

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  14. Misc I Q.14
    The 2 vectors j^+k^\hat{j} + \hat{k} and 3i^−j^+4k^3\hat{i} - \hat{j} + 4\hat{k} represents the two sides AB and AC, respectively of a △ABC\triangle ABC. The length of the median through A is

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  15. Misc I Q.15
    If a⃗\vec{a} and b⃗\vec{b} are unit vectors, then what is the angle between a⃗\vec{a} and b⃗\vec{b} for 3a⃗−b⃗\sqrt{3}\vec{a} - \vec{b} to be a unit vector?

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  16. Misc I Q.16
    If θ\theta be the angle between any two vectors a⃗\vec{a} and b⃗\vec{b}, then ∣a⃗⋅b⃗∣=∣a⃗×b⃗∣|\vec{a}\cdot\vec{b}| = |\vec{a}\times\vec{b}|, when θ\theta is equal to

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  17. Misc I Q.17
    The value of i^⋅(j^×k^)+j^⋅(i^×k^)+k^⋅(i^×j^)\hat{i}\cdot(\hat{j}\times\hat{k}) + \hat{j}\cdot(\hat{i}\times\hat{k}) + \hat{k}\cdot(\hat{i}\times\hat{j})

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  18. Misc I Q.18
    Let a, b, c be distinct non-negative numbers. If the vectors ai^+aj^+ck^a\hat{i}+a\hat{j}+c\hat{k}, i^+k^\hat{i}+\hat{k} and ci^+cj^+bk^c\hat{i}+c\hat{j}+b\hat{k} lie in a plane, then c is

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  19. Misc I Q.19
    Let a⃗=i^−j^\vec{a} = \hat{i}-\hat{j}, b⃗=j^−k^\vec{b} = \hat{j}-\hat{k}, c⃗=k^−i^\vec{c} = \hat{k}-\hat{i}. If d⃗\vec{d} is a unit vector such that a⃗⋅d⃗=0=[b⃗ c⃗ d⃗]\vec{a}\cdot\vec{d}=0=[\vec{b}\ \vec{c}\ \vec{d}], then d⃗\vec{d} equals.

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  20. Misc I Q.20
    If a⃗ b⃗ c⃗\vec{a}\ \vec{b}\ \vec{c} are non coplanar unit vectors such that a⃗×(b⃗×c⃗)=(b⃗+c⃗)2\vec{a}\times(\vec{b}\times\vec{c}) = \dfrac{(\vec{b}+\vec{c})}{\sqrt{2}} then the angle between a⃗\vec{a} and b⃗\vec{b} is

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Solve the following

Practice · 67
  1. ABCD is a trapezium with AB parallel to DC and DC = 3AB. M is the mid-point of DC, AB→=p⃗\overrightarrow{AB} = \vec{p} and BC→=q⃗\overrightarrow{BC} = \vec{q}. Find in terms of p⃗\vec{p} and q⃗\vec{q}.
    Misc II Q.1 i)
    AM→\overrightarrow{AM}
  2. Misc II Q.1 ii)
    BD→\overrightarrow{BD}
  3. Misc II Q.1 iii)
    MB→\overrightarrow{MB}
  4. Misc II Q.1 iv)
    DA→\overrightarrow{DA}
  5. Misc II Q.2
    The points A, B and C have position vectors a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} respectively. The point P is midpoint of AB. Find in terms of a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} the vector PC→\overrightarrow{PC}.
  6. Misc II Q.3
    In a pentagon ABCDE, show that AB→+AE→+BC→+DC→+ED→=2AC→\overrightarrow{AB} + \overrightarrow{AE} + \overrightarrow{BC} + \overrightarrow{DC} + \overrightarrow{ED} = 2\overrightarrow{AC}.
  7. Misc II Q.4
    If in parallelogram ABCD, diagonal vectors are AC→=2i^+3j^+4k^\overrightarrow{AC} = 2\hat{i}+3\hat{j}+4\hat{k} and BD→=−6i^+7j^−2k^\overrightarrow{BD} = -6\hat{i}+7\hat{j}-2\hat{k}, then find the adjacent side vectors AB→\overrightarrow{AB} and AD→\overrightarrow{AD}.
  8. Misc II Q.5
    If two sides of a triangle taken in the same order represented by vectors i^+2j^\hat{i}+2\hat{j} and i^+k^\hat{i}+\hat{k}, then find the length of the third side.
  9. Misc II Q.6
    If ∣a⃗∣=∣b⃗∣=1|\vec{a}| = |\vec{b}| = 1, a⃗⋅b⃗=0\vec{a}\cdot\vec{b}=0 and a⃗+b⃗+c⃗=0⃗\vec{a}+\vec{b}+\vec{c}=\vec{0} then find ∣c⃗∣|\vec{c}|.
  10. Find the lengths of the sides of the triangle and also determine the type of a triangle.
    Misc II Q.7 i)
    A(2,−1,0)A(2,-1,0), B(4,1,1)B(4,1,1), C(4,−5,4)C(4,-5,4)
  11. Misc II Q.7 ii)
    L(3,−2,−3)L(3,-2,-3), M(7,0,1)M(7,0,1), N(1,2,1)N(1,2,1)
  12. Find the component form of a⃗\vec{a} if
    Misc II Q.8 i)
    It lies in YZ plane and makes 60∘60^\circ with positive Y-axis and ∣a⃗∣=4|\vec{a}| = 4
  13. Misc II Q.8 ii)
    It lies in XZ plane and makes 45∘45^\circ with positive Z-axis and ∣a⃗∣=10|\vec{a}| = 10
  14. Misc II Q.9
    Two sides of a parallelogram are 3i^+4j^−5k^3\hat{i}+4\hat{j}-5\hat{k} and −2j^+7k^-2\hat{j}+7\hat{k}. Find the unit vectors parallel to the diagonals.
  15. Misc II Q.10
    If D, E, F are the mid-points of the sides BC, CA, AB of a triangle ABC, prove that AD→+BE→+CF→=0⃗\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = \vec{0}.
  16. Misc II Q.11
    Find the unit vectors that are parallel to the tangent line to the parabola y=x2y = x^2 at the point (2,4)(2, 4).
  17. Misc II Q.12
    Express the vector i^+4j^−4k^\hat{i}+4\hat{j}-4\hat{k} as a linear combination of the vectors 2i^−j^+3k^2\hat{i}-\hat{j}+3\hat{k}, i^−2j^+4k^\hat{i}-2\hat{j}+4\hat{k} and −i^+3j^−5k^-\hat{i}+3\hat{j}-5\hat{k}.
  18. Misc II Q.13
    If OA→=a⃗\overrightarrow{OA} = \vec{a} and OB→=b⃗\overrightarrow{OB} = \vec{b} then show that vector along the angle bisector of angle AOB is given by d⃗=λ(a⃗∣a⃗∣+b⃗∣b⃗∣)\vec{d} = \lambda\left(\dfrac{\vec{a}}{|\vec{a}|}+\dfrac{\vec{b}}{|\vec{b}|}\right), where λ\lambda is a real number.
  19. Misc II Q.14
    The position vectors of three consecutive vertices of a parallelogram are i^+j^+k^\hat{i}+\hat{j}+\hat{k}, i^+3j^+5k^\hat{i}+3\hat{j}+5\hat{k} and 7i^+9j^+11k^7\hat{i}+9\hat{j}+11\hat{k}. Find the position vector of the fourth vertex.
  20. Misc II Q.15
    A point P with p.v. −14i^+39j^+28k^5\dfrac{-14\hat{i}+39\hat{j}+28\hat{k}}{5} divides the line joining A(−1,6,5)A(-1, 6, 5) and B internally in the ratio 3:2 then find the point B.
  21. Misc II Q.16
    Prove that the sum of the three vectors determined by the medians of a triangle directed from the vertices is zero.
  22. Misc II Q.17
    ABCD is a parallelogram E, F are the mid points of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.
  23. Misc II Q.18
    If ABC is a triangle whose orthocenter is P and the circumcenter is Q, then prove that PA→+PC→+PB→=2PQ→\overrightarrow{PA} + \overrightarrow{PC} + \overrightarrow{PB} = 2\overrightarrow{PQ}.
  24. Misc II Q.19
    If P is orthocenter, Q is circumcenter and G is centroid of triangle ABC, then prove that QP→=3QG→\overrightarrow{QP} = 3\overrightarrow{QG}.
  25. Misc II Q.20
    In triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, determine the ratio OP : PD using vector methods.
  26. Misc II Q.21
    Dot-product of a vector with vectors 3i^−5k^3\hat{i}-5\hat{k}, 2i^+7j^2\hat{i}+7\hat{j} and i^+j^+k^\hat{i}+\hat{j}+\hat{k} are respectively -1, 6 and 5. Find the vector.
  27. Misc II Q.22
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are unit vectors such that a⃗+b⃗+c⃗=0⃗\vec{a}+\vec{b}+\vec{c}=\vec{0}, then find the value of a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}.
  28. Misc II Q.23
    If a parallelogram is constructed on the vectors a⃗=3p⃗−q⃗\vec{a} = 3\vec{p} - \vec{q}, b⃗=p⃗+3q⃗\vec{b} = \vec{p} + 3\vec{q} and ∣p⃗∣=∣q⃗∣=2|\vec{p}| = |\vec{q}| = 2 and angle between p⃗\vec{p} and q⃗\vec{q} is π/3\pi/3 show that the ratio of the lengths of the sides is 7:13\sqrt{7} : \sqrt{13}.
  29. Misc II Q.24
    Express the vector a⃗=5i^−2j^+5k^\vec{a} = 5\hat{i}-2\hat{j}+5\hat{k} as a sum of two vectors such that one is parallel to the vector b⃗=3i^+k^\vec{b} = 3\hat{i}+\hat{k} and other is perpendicular to b⃗\vec{b}.
  30. Misc II Q.25
    Find two unit vectors each of which makes equal angles with u⃗,v⃗\vec{u}, \vec{v} and w⃗\vec{w}. u⃗=2i^+j^−2k^\vec{u} = 2\hat{i}+\hat{j}-2\hat{k}, v⃗=i^+2j^−2k^\vec{v} = \hat{i}+2\hat{j}-2\hat{k} and w⃗=2i^−2j^+k^\vec{w} = 2\hat{i}-2\hat{j}+\hat{k}.
  31. Misc II Q.26
    Find the acute angles between the curves at their points of intersection. y=x2y = x^2, y=x3y = x^3
  32. Find the direction cosines and direction angles of the vector.
    Misc II Q.27 i)
    2i^+j^+2k^2\hat{i}+\hat{j}+2\hat{k}
  33. Misc II Q.27 ii)
    12i^+j^+k^\dfrac{1}{2}\hat{i}+\hat{j}+\hat{k}
  34. Misc II Q.28
    Let b⃗=4i^+3j^\vec{b} = 4\hat{i}+3\hat{j} and c⃗\vec{c} be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along b⃗\vec{b} and c⃗\vec{c}, respectively.
  35. Misc II Q.29
    Show that no line in space can make angles π/6\pi/6 and π/4\pi/4 with X-axis and Y-axis respectively.
  36. Misc II Q.30
    Find the angle between the lines whose direction cosines are given by the equation 6mn−2nl+5lm=06mn-2nl+5lm=0, 3l+m+5n=03l+m+5n=0.
  37. Misc II Q.31
    If Q is the foot of the perpendicular from P(2,4,3)P(2,4,3) on the line joining the points A(1,2,4)A(1,2,4) and B(3,4,5)B(3,4,5), find coordinates of Q.
  38. Misc II Q.32
    Show that the area of triangle ABC, the position vectors of whose vertices are a, b and c is 12[a⃗×b⃗+b⃗×c⃗+c⃗×a⃗]\dfrac{1}{2}\left[\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}\right].
  39. Misc II Q.33
    Find a unit vector perpendicular to the plane containing the point (a,0,0)(a, 0, 0), (0,b,0)(0, b, 0), and (0,0,c)(0, 0, c). What is the area of the triangle with these vertices?
  40. State whether each expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar.
    Misc II Q.34 (a)
    a⃗⋅(b⃗×c⃗)\vec{a}\cdot(\vec{b}\times\vec{c})
  41. Misc II Q.34 (b)
    a⃗×(b⃗⋅c⃗)\vec{a}\times(\vec{b}\cdot\vec{c})
  42. Misc II Q.34 (c)
    a⃗×(b⃗×c⃗)\vec{a}\times(\vec{b}\times\vec{c})
  43. Misc II Q.34 (d)
    a⃗⋅(b⃗⋅c⃗)\vec{a}\cdot(\vec{b}\cdot\vec{c})
  44. Misc II Q.34 (e)
    (a⃗⋅b⃗)×(c⃗⋅d⃗)(\vec{a}\cdot\vec{b})\times(\vec{c}\cdot\vec{d})
  45. Misc II Q.34 (f)
    (a⃗×b⃗)⋅(c⃗×d⃗)(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})
  46. Misc II Q.34 (g)
    (a⃗⋅b⃗)⋅c⃗(\vec{a}\cdot\vec{b})\cdot\vec{c}
  47. Misc II Q.34 (h)
    (a⃗⋅b⃗)c⃗(\vec{a}\cdot\vec{b})\vec{c}
  48. Misc II Q.34 (i)
    (∣a⃗∣)(b⃗⋅c⃗)(|\vec{a}|)(\vec{b}\cdot\vec{c})
  49. Misc II Q.34 (j)
    a⃗⋅(b⃗+c⃗)\vec{a}\cdot(\vec{b}+\vec{c})
  50. Misc II Q.34 (k)
    a⃗⋅b⃗+c⃗\vec{a}\cdot\vec{b}+\vec{c}
  51. Misc II Q.34 (l)
    ∣a⃗∣⋅(b⃗+c⃗)|\vec{a}|\cdot(\vec{b}+\vec{c})
  52. Misc II Q.35
    Show that, for any vectors a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c}: (a⃗+b⃗+c⃗)×c⃗+(a⃗+b⃗+c⃗)×b⃗+(b⃗+c⃗)×a⃗=2a⃗×c⃗(\vec{a}+\vec{b}+\vec{c})\times\vec{c}+(\vec{a}+\vec{b}+\vec{c})\times\vec{b}+(\vec{b}+\vec{c})\times\vec{a}=2\vec{a}\times\vec{c}.
  53. Suppose that a⃗≠0⃗\vec{a} \ne \vec{0}.
    Misc II Q.36 (a)
    If a⃗⋅b⃗=a⃗⋅c⃗\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} then is b⃗=c⃗\vec{b} = \vec{c}?
  54. Misc II Q.36 (b)
    If a⃗×b⃗=a⃗×c⃗\vec{a}\times\vec{b} = \vec{a}\times\vec{c} then is b⃗=c⃗\vec{b} = \vec{c}?
  55. Misc II Q.36 (c)
    If a⃗⋅b⃗=a⃗⋅c⃗\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} and a⃗×b⃗=a⃗×c⃗\vec{a}\times\vec{b} = \vec{a}\times\vec{c} then is b⃗=c⃗\vec{b} = \vec{c}?
  56. Misc II Q.37
    If A(3,2,−1)A(3,2,-1), B(−2,2,−3)B(-2,2,-3), C(3,5,−2)C(3,5,-2), D(−2,5,−4)D(-2,5,-4) then (i) verify that the points are the vertices of a parallelogram and (ii) find its area.
  57. Misc II Q.38
    Let A, B, C, D be any four points in space. Prove that ∣AB→×CD→+BC→×AD→+CA→×BD→∣=4(area of △ABC)|\overrightarrow{AB}\times\overrightarrow{CD}+\overrightarrow{BC}\times\overrightarrow{AD}+\overrightarrow{CA}\times\overrightarrow{BD}| = 4(\text{area of } \triangle ABC).
  58. Misc II Q.39
    Let a^,b^,c^\hat{a}, \hat{b}, \hat{c} be unit vectors such that a^⋅b^=a^⋅c^=0\hat{a}\cdot\hat{b} = \hat{a}\cdot\hat{c} = 0 and the angle between b^\hat{b} and c^\hat{c} be π/6\pi/6. Prove that a^=±2(b^×c^)\hat{a} = \pm 2(\hat{b}\times\hat{c}).
  59. Misc II Q.40
    Find the value of 'a' so that the volume of parallelopiped formed by i^+aj^+k^\hat{i}+a\hat{j}+\hat{k}, j^+ak^\hat{j}+a\hat{k} and ai^+k^a\hat{i}+\hat{k} becomes minimum.
  60. Misc II Q.41
    Find the volume of the parallelepiped spanned by the diagonals of the three faces of a cube of side aa that meet at one vertex of the cube.
  61. Misc II Q.42
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three non-coplanar vectors, then show that a⃗⋅(b⃗×c⃗)(c⃗×a⃗)⋅b⃗+b⃗⋅(a⃗×c⃗)(c⃗×a⃗)⋅b⃗=0\dfrac{\vec{a}\cdot(\vec{b}\times\vec{c})}{(\vec{c}\times\vec{a})\cdot\vec{b}} + \dfrac{\vec{b}\cdot(\vec{a}\times\vec{c})}{(\vec{c}\times\vec{a})\cdot\vec{b}} = 0.
  62. Misc II Q.43
    Prove that (a⃗×b⃗)⋅(c⃗×d⃗)=∣a⃗⋅c⃗b⃗⋅c⃗a⃗⋅d⃗b⃗⋅d⃗∣(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = \begin{vmatrix}\vec{a}\cdot\vec{c} & \vec{b}\cdot\vec{c}\\ \vec{a}\cdot\vec{d} & \vec{b}\cdot\vec{d}\end{vmatrix}.
  63. Misc II Q.44
    Find the volume of the parallelopiped whose coterminus edges are represented by the vector j^+k^\hat{j}+\hat{k}, i^+k^\hat{i}+\hat{k} and i^+j^\hat{i}+\hat{j}. Also find volume of tetrahedron having these coterminous edges.
  64. Misc II Q.45
    Using properties of scalar triple product, prove that [a⃗+b⃗  b⃗+c⃗  c⃗+a⃗]=2[a⃗ b⃗ c⃗][\vec{a}+\vec{b}\ \ \vec{b}+\vec{c}\ \ \vec{c}+\vec{a}] = 2[\vec{a}\ \vec{b}\ \vec{c}].
  65. Misc II Q.46
    If four points A(a⃗)A(\vec{a}), B(b⃗)B(\vec{b}), C(c⃗)C(\vec{c}) and D(d⃗)D(\vec{d}) are coplanar then show that [a⃗ b⃗ d⃗]+[b⃗ c⃗ d⃗]+[c⃗ a⃗ d⃗]=[a⃗ b⃗ c⃗][\vec{a}\ \vec{b}\ \vec{d}] + [\vec{b}\ \vec{c}\ \vec{d}] + [\vec{c}\ \vec{a}\ \vec{d}] = [\vec{a}\ \vec{b}\ \vec{c}].
  66. Misc II Q.47
    If a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are three non coplanar vectors, then (a⃗+b⃗+c⃗)⋅[(a⃗+b⃗)×(a⃗+c⃗)]=−[a⃗ b⃗ c⃗](\vec{a}+\vec{b}+\vec{c})\cdot\left[(\vec{a}+\vec{b})\times(\vec{a}+\vec{c})\right] = -[\vec{a}\ \vec{b}\ \vec{c}].
  67. Misc II Q.48
    If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair are also perpendicular.