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Mathematics · Textbook solutions

Vector Algebra

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 125 questions

10.3 Types of Vectors

29 q

Solved Examples

Worked · 10
  1. 10.1 Eg.1
    Represent graphically a displacement of 40 km, 30∘30^\circ west of south.
  2. Classify the following measures as scalars and vectors.
    10.1 Eg.2(i)
    5 seconds
  3. 10.1 Eg.2(ii)
    1000 cm3\text{cm}^{3}
  4. 10.1 Eg.2(iii)
    10 Newton
  5. 10.1 Eg.2(iv)
    30 km/hr
  6. 10.1 Eg.2(v)
    10 g/cm3\text{g/cm}^{3}
  7. 10.1 Eg.2(vi)
    20 m/s towards north
  8. In Fig 10.5, which of the vectors are:
    10.1 Eg.3(i)
    Collinear
  9. 10.1 Eg.3(ii)
    Equal
  10. 10.1 Eg.3(iii)
    Coinitial

Exercise 10.1

Practice · 19
  1. Ex 10.1 Q1
    Represent graphically a displacement of 40 km, 30∘30^\circ east of north.
  2. Classify the following measures as scalars and vectors.
    Ex 10.1 Q2(i)
    10 kg
  3. Ex 10.1 Q2(ii)
    2 meters north-west
  4. Ex 10.1 Q2(iii)
    40∘40^\circ
  5. Ex 10.1 Q2(iv)
    40 watt
  6. Ex 10.1 Q2(v)
    10−1910^{-19} coulomb
  7. Ex 10.1 Q2(vi)
    20 m/s2\text{m/s}^{2}
  8. Classify the following as scalar and vector quantities.
    Ex 10.1 Q3(i)
    time period
  9. Ex 10.1 Q3(ii)
    distance
  10. Ex 10.1 Q3(iii)
    force
  11. Ex 10.1 Q3(iv)
    velocity
  12. Ex 10.1 Q3(v)
    work done
  13. In Fig 10.6 (a square), identify the following vectors.
    Ex 10.1 Q4(i)
    Coinitial
  14. Ex 10.1 Q4(ii)
    Equal
  15. Ex 10.1 Q4(iii)
    Collinear but not equal
  16. Answer the following as true or false.
    Ex 10.1 Q5(i)
    a⃗\vec{a} and −a⃗-\vec{a} are collinear.
  17. Ex 10.1 Q5(ii)
    Two collinear vectors are always equal in magnitude.
  18. Ex 10.1 Q5(iii)
    Two vectors having same magnitude are collinear.
  19. Ex 10.1 Q5(iv)
    Two collinear vectors having the same magnitude are equal.

10.4-10.5 Addition, Components and Section Formula

29 q

Solved Examples

Worked · 9
  1. 10.2 Eg.4
    Find the values of xx, yy and zz so that the vectors a⃗=xi^+2j^+zk^\vec{a} = x\hat{i} + 2\hat{j} + z\hat{k} and b⃗=2i^+yj^+k^\vec{b} = 2\hat{i} + y\hat{j} + \hat{k} are equal.
  2. 10.2 Eg.5
    Let a⃗=i^+2j^\vec{a} = \hat{i} + 2\hat{j} and b⃗=2i^+j^\vec{b} = 2\hat{i} + \hat{j}. Is ∣a⃗∣=∣b⃗∣|\vec{a}| = |\vec{b}|? Are the vectors a⃗\vec{a} and b⃗\vec{b} equal?
  3. 10.2 Eg.6
    Find unit vector in the direction of vector a⃗=2i^+3j^+k^\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}.
  4. 10.2 Eg.7
    Find a vector in the direction of vector a⃗=i^−2j^\vec{a} = \hat{i} - 2\hat{j} that has magnitude 7 units.
  5. 10.2 Eg.8
    Find the unit vector in the direction of the sum of the vectors, a⃗=2i^+2j^−5k^\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k} and b⃗=2i^+j^+3k^\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}.
  6. 10.2 Eg.9
    Write the direction ratio's of the vector a⃗=i^+j^−2k^\vec{a} = \hat{i} + \hat{j} - 2\hat{k} and hence calculate its direction cosines.
  7. 10.2 Eg.10
    Find the vector joining the points P(2,3,0)\mathrm{P}(2, 3, 0) and Q(−1,−2,−4)\mathrm{Q}(-1, -2, -4) directed from P to Q.
  8. 10.2 Eg.11
    Consider two points P and Q with position vectors OP→=3a⃗−2b⃗\overrightarrow{\mathrm{OP}} = 3\vec{a} - 2\vec{b} and OQ→=a⃗+b⃗\overrightarrow{\mathrm{OQ}} = \vec{a} + \vec{b}. Find the position vector of a point R which divides the line joining P and Q in the ratio 2:1, (i) internally, and (ii) externally.
  9. 10.2 Eg.12
    Show that the points A(2i^−j^+k^)\mathrm{A}(2\hat{i} - \hat{j} + \hat{k}), B(i^−3j^−5k^)\mathrm{B}(\hat{i} - 3\hat{j} - 5\hat{k}), C(3i^−4j−4k^)\mathrm{C}(3\hat{i} - 4j - 4\hat{k}) are the vertices of a right angled triangle.

Exercise 10.2

Practice · 20
  1. Ex 10.2 Q1
    Compute the magnitude of the following vectors: a⃗=i^+j^+k;b⃗=2i^−7j^−3k^;c⃗=13i^+13j^−13k^\vec{a} = \hat{i} + \hat{j} + k;\quad \vec{b} = 2\hat{i} - 7\hat{j} - 3\hat{k};\quad \vec{c} = \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} - \frac{1}{\sqrt{3}}\hat{k}
  2. Ex 10.2 Q2
    Write two different vectors having same magnitude.
  3. Ex 10.2 Q3
    Write two different vectors having same direction.
  4. Ex 10.2 Q4
    Find the values of xx and yy so that the vectors 2i^+3j^2\hat{i} + 3\hat{j} and xi^+yj^x\hat{i} + y\hat{j} are equal.
  5. Ex 10.2 Q5
    Find the scalar and vector components of the vector with initial point (2,1)(2, 1) and terminal point (−5,7)(-5, 7).
  6. Ex 10.2 Q6
    Find the sum of the vectors a⃗=i^−2j^+k^\vec{a} = \hat{i} - 2\hat{j} + \hat{k}, b⃗=−2i^+4j^+5k^\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k} and c⃗=i^−6j^−7k^\vec{c} = \hat{i} - 6\hat{j} - 7\hat{k}.
  7. Ex 10.2 Q7
    Find the unit vector in the direction of the vector a⃗=i^+j^+2k^\vec{a} = \hat{i} + \hat{j} + 2\hat{k}.
  8. Ex 10.2 Q8
    Find the unit vector in the direction of vector PQ→\overrightarrow{\mathrm{PQ}}, where P and Q are the points (1,2,3)(1, 2, 3) and (4,5,6)(4, 5, 6), respectively.
  9. Ex 10.2 Q9
    For given vectors, a⃗=2i^−j^+2k^\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} and b⃗=−i^+j^−k^\vec{b} = -\hat{i} + \hat{j} - \hat{k}, find the unit vector in the direction of the vector a⃗+b⃗\vec{a} + \vec{b}.
  10. Ex 10.2 Q10
    Find a vector in the direction of vector 5i^−j^+2k^5\hat{i} - \hat{j} + 2\hat{k} which has magnitude 8 units.
  11. Ex 10.2 Q11
    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 collinear.
  12. Ex 10.2 Q12
    Find the direction cosines of the vector i^+2j^+3k^\hat{i} + 2\hat{j} + 3\hat{k}.
  13. Ex 10.2 Q13
    Find the direction cosines of the vector joining the points A(1,2,−3)\mathrm{A}(1, 2, -3) and B(−1,−2,1)\mathrm{B}(-1, -2, 1), directed from A to B.
  14. Ex 10.2 Q14
    Show that the vector i^+j^+k^\hat{i} + \hat{j} + \hat{k} is equally inclined to the axes OX, OY and OZ.
  15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are i^+2j^−k^\hat{i} + 2\hat{j} - \hat{k} and −i^+j^+k^-\hat{i} + \hat{j} + \hat{k} respectively, in the ratio 2 : 1
    Ex 10.2 Q15(i)
    internally
  16. Ex 10.2 Q15(ii)
    externally
  17. Ex 10.2 Q16
    Find the position vector of the mid point of the vector joining the points P(2,3,4)\mathrm{P}(2, 3, 4) and Q(4,1,−2)\mathrm{Q}(4, 1, -2).
  18. Ex 10.2 Q17
    Show that the points A, B and C with position vectors, a⃗=3i^−4j^−4k^\vec{a} = 3\hat{i} - 4\hat{j} - 4\hat{k}, b⃗=2i^−j^+k^\vec{b} = 2\hat{i} - \hat{j} + \hat{k} and c⃗=i^−3j^−5k^\vec{c} = \hat{i} - 3\hat{j} - 5\hat{k}, respectively form the vertices of a right angled triangle.
  19. Ex 10.2 Q18
    In triangle ABC (Fig 10.18), which of the following is not true:

    Tap an option to check your answer.

  20. Ex 10.2 Q19
    If a⃗\vec{a} and b⃗\vec{b} are two collinear vectors, then which of the following are incorrect:

    Tap an option to check your answer.

10.6.1-10.6.2 Scalar (Dot) Product and Projection

27 q

Solved Examples

Worked · 9
  1. 10.3 Eg.13
    Find the angle between two vectors a⃗\vec{a} and b⃗\vec{b} with magnitudes 1 and 2 respectively and when a⃗⋅b⃗=1\vec{a} \cdot \vec{b} = 1.
  2. 10.3 Eg.14
    Find angle θ\theta between the vectors a⃗=i^+j^−k^\vec{a} = \hat{i} + \hat{j} - \hat{k} and b⃗=i^−j^+k^\vec{b} = \hat{i} - \hat{j} + \hat{k}.
  3. 10.3 Eg.15
    If a⃗=5i^−j^−3k^\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k} and b⃗=i^+3j^−5k^\vec{b} = \hat{i} + 3\hat{j} - 5\hat{k}, then show that the vectors a⃗+b⃗\vec{a} + \vec{b} and a⃗−b⃗\vec{a} - \vec{b} are perpendicular.
  4. 10.3 Eg.16
    Find the projection of the vector a⃗=2i^+3j^+2k^\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k} on the vector b⃗=i^+2j^+k^\vec{b} = \hat{i} + 2\hat{j} + \hat{k}.
  5. 10.3 Eg.17
    Find ∣a⃗−b⃗∣|\vec{a} - \vec{b}|, if two vectors a⃗\vec{a} and b⃗\vec{b} are such that ∣a⃗∣=2|\vec{a}| = 2, ∣b⃗∣=3|\vec{b}| = 3 and a⃗⋅b⃗=4\vec{a} \cdot \vec{b} = 4.
  6. 10.3 Eg.18
    If a⃗\vec{a} is a unit vector and (x⃗−a⃗)⋅(x⃗+a⃗)=8(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 8, then find ∣x⃗∣|\vec{x}|.
  7. 10.3 Eg.19
    For any two vectors a⃗\vec{a} and b⃗\vec{b}, we always have ∣a⃗⋅b⃗∣≤∣a⃗∣∣b⃗∣|\vec{a} \cdot \vec{b}| \le |\vec{a}||\vec{b}| (Cauchy-Schwartz inequality).
  8. 10.3 Eg.20
    For any two vectors a⃗\vec{a} and b⃗\vec{b}, we always have ∣a⃗+b⃗∣≤∣a⃗∣+∣b⃗∣|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}| (triangle inequality).
  9. 10.3 Eg.21
    Show that the points A(−2i^+3j^+5k^)\mathrm{A}(-2\hat{i} + 3\hat{j} + 5\hat{k}), B(i^+2j^+3k^)\mathrm{B}(\hat{i} + 2\hat{j} + 3\hat{k}) and C(7i^−k^)\mathrm{C}(7\hat{i} - \hat{k}) are collinear.

Exercise 10.3

Practice · 18
  1. Ex 10.3 Q1
    Find the angle between two vectors a⃗\vec{a} and b⃗\vec{b} with magnitudes 3\sqrt{3} and 2, respectively having a⃗⋅b⃗=6\vec{a} \cdot \vec{b} = \sqrt{6}.
  2. Ex 10.3 Q2
    Find the angle between the vectors i^−2j^+3k^\hat{i} - 2\hat{j} + 3\hat{k} and 3i^−2j^+k^3\hat{i} - 2\hat{j} + \hat{k}
  3. Ex 10.3 Q3
    Find the projection of the vector i^−j^\hat{i} - \hat{j} on the vector i^+j^\hat{i} + \hat{j}.
  4. Ex 10.3 Q4
    Find the projection of the vector i^+3j^+7k^\hat{i} + 3\hat{j} + 7\hat{k} on the vector 7i^−j^+8k^7\hat{i} - \hat{j} + 8\hat{k}.
  5. Ex 10.3 Q5
    Show that each of the given three vectors is a unit vector: 17(2i^+3j^+6k^)\frac{1}{7}(2\hat{i} + 3\hat{j} + 6\hat{k}), 17(3i^−6j^+2k^)\frac{1}{7}(3\hat{i} - 6\hat{j} + 2\hat{k}), 17(6i^+2j^−3k^)\frac{1}{7}(6\hat{i} + 2\hat{j} - 3\hat{k}) Also, show that they are mutually perpendicular to each other.
  6. Ex 10.3 Q6
    Find ∣a⃗∣|\vec{a}| and ∣b⃗∣|\vec{b}|, if (a⃗+b⃗)⋅(a⃗−b⃗)=8(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 8 and ∣a⃗∣=8∣b⃗∣|\vec{a}| = 8|\vec{b}|.
  7. Ex 10.3 Q7
    Evaluate the product (3a⃗−5b⃗)⋅(2a⃗+7b⃗)(3\vec{a} - 5\vec{b}) \cdot (2\vec{a} + 7\vec{b}).
  8. Ex 10.3 Q8
    Find the magnitude of two vectors a⃗\vec{a} and b⃗\vec{b}, having the same magnitude and such that the angle between them is 60∘60^\circ and their scalar product is 12\frac{1}{2}.
  9. Ex 10.3 Q9
    Find ∣x⃗∣|\vec{x}|, if for a unit vector a⃗\vec{a}, (x⃗−a⃗)⋅(x⃗+a⃗)=12(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 12.
  10. Ex 10.3 Q10
    If a⃗=2i^+2j^+3k^\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}, b⃗=−i^+2j^+k^\vec{b} = -\hat{i} + 2\hat{j} + \hat{k} and c⃗=3i^+j^\vec{c} = 3\hat{i} + \hat{j} are such that a⃗+λb⃗\vec{a} + \lambda\vec{b} is perpendicular to c⃗\vec{c}, then find the value of λ\lambda.
  11. Ex 10.3 Q11
    Show that ∣a⃗∣b⃗+∣b⃗∣a⃗|\vec{a}|\vec{b} + |\vec{b}|\vec{a} is perpendicular to ∣a⃗∣b⃗−∣b⃗∣a⃗|\vec{a}|\vec{b} - |\vec{b}|\vec{a}, for any two nonzero vectors a⃗\vec{a} and b⃗\vec{b}.
  12. Ex 10.3 Q12
    If a⃗⋅a⃗=0\vec{a} \cdot \vec{a} = 0 and a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0, then what can be concluded about the vector b⃗\vec{b}?
  13. Ex 10.3 Q13
    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}, 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}.
  14. Ex 10.3 Q14
    If either vector a⃗=0⃗\vec{a} = \vec{0} or b⃗=0⃗\vec{b} = \vec{0}, then a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0. But the converse need not be true. Justify your answer with an example.
  15. Ex 10.3 Q15
    If the vertices A, B, C of a triangle ABC are (1,2,3)(1, 2, 3), (−1,0,0)(-1, 0, 0), (0,1,2)(0, 1, 2), respectively, then find ∠ABC\angle \mathrm{ABC}. [∠ABC\angle \mathrm{ABC} is the angle between the vectors BA→\overrightarrow{\mathrm{BA}} and BC→\overrightarrow{\mathrm{BC}}].
  16. Ex 10.3 Q16
    Show that the points A(1,2,7)\mathrm{A}(1, 2, 7), B(2,6,3)\mathrm{B}(2, 6, 3) and C(3,10,−1)\mathrm{C}(3, 10, -1) are collinear.
  17. Ex 10.3 Q17
    Show that the vectors 2i^−j^+k^2\hat{i} - \hat{j} + \hat{k}, i^−3j^−5k^\hat{i} - 3\hat{j} - 5\hat{k} and 3i^−4j^−4k^3\hat{i} - 4\hat{j} - 4\hat{k} form the vertices of a right angled triangle.
  18. Ex 10.3 Q18
    If a⃗\vec{a} is a nonzero vector of magnitude 'aa' and λ\lambda a nonzero scalar, then λa⃗\lambda\vec{a} is unit vector if

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10.6.3 Vector (Cross) Product of Two Vectors

16 q

Solved Examples

Worked · 4
  1. 10.4 Eg.22
    Find ∣a⃗×b⃗∣|\vec{a} \times \vec{b}|, if a⃗=2i^+j^+3k^\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} and b⃗=3i^+5j^−2k^\vec{b} = 3\hat{i} + 5\hat{j} - 2\hat{k}.
  2. 10.4 Eg.23
    Find a unit vector perpendicular to each of the vectors (a⃗+b⃗)(\vec{a} + \vec{b}) and (a⃗−b⃗)(\vec{a} - \vec{b}), where a⃗=i^+j^+k^\vec{a} = \hat{i} + \hat{j} + \hat{k}, b⃗=i^+2j^+3k^\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}.
  3. 10.4 Eg.24
    Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices.
  4. 10.4 Eg.25
    Find the area of a parallelogram whose adjacent sides are given by the vectors a⃗=3i^+j^+4k^\vec{a} = 3\hat{i} + \hat{j} + 4\hat{k} and b⃗=i^−j^+k^\vec{b} = \hat{i} - \hat{j} + \hat{k}.

Exercise 10.4

Practice · 12
  1. Ex 10.4 Q1
    Find ∣a⃗×b⃗∣|\vec{a} \times \vec{b}|, if a⃗=i^−7j^+7k^\vec{a} = \hat{i} - 7\hat{j} + 7\hat{k} and b⃗=3i^−2j^+2k^\vec{b} = 3\hat{i} - 2\hat{j} + 2\hat{k}.
  2. Ex 10.4 Q2
    Find a unit vector perpendicular to each of the vector a⃗+b⃗\vec{a} + \vec{b} and a⃗−b⃗\vec{a} - \vec{b}, where a⃗=3i^+2j^+2k^\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k} and b⃗=i^+2j^−2k^\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}.
  3. Ex 10.4 Q3
    If a unit vector a⃗\vec{a} makes angles π3\frac{\pi}{3} with i^\hat{i}, π4\frac{\pi}{4} with j^\hat{j} and an acute angle θ\theta with k^\hat{k}, then find θ\theta and hence, the components of a⃗\vec{a}.
  4. Ex 10.4 Q4
    Show that (a⃗−b⃗)×(a⃗+b⃗)=2(a⃗×b⃗)(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = 2(\vec{a} \times \vec{b}).
  5. Ex 10.4 Q5
    Find λ\lambda and μ\mu if (2i^+6j^+27k^)×(i^+λj^+μk^)=0⃗(2\hat{i} + 6\hat{j} + 27\hat{k}) \times (\hat{i} + \lambda\hat{j} + \mu\hat{k}) = \vec{0}.
  6. Ex 10.4 Q6
    Given that a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0 and a⃗×b⃗=0⃗\vec{a} \times \vec{b} = \vec{0}. What can you conclude about the vectors a⃗\vec{a} and b⃗\vec{b}?
  7. Ex 10.4 Q7
    Let the vectors a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} be given as a1i^+a2j^+a3k^a_1\hat{i} + a_2\hat{j} + a_3\hat{k}, b1i^+b2j^+b3k^b_1\hat{i} + b_2\hat{j} + b_3\hat{k}, c1i^+c2j^+c3k^c_1\hat{i} + c_2\hat{j} + c_3\hat{k}. Then show 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}.
  8. Ex 10.4 Q8
    If either a⃗=0⃗\vec{a} = \vec{0} or b⃗=0⃗\vec{b} = \vec{0}, then a⃗×b⃗=0⃗\vec{a} \times \vec{b} = \vec{0}. Is the converse true? Justify your answer with an example.
  9. Ex 10.4 Q9
    Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
  10. Ex 10.4 Q10
    Find the area of the parallelogram whose adjacent sides are determined by the vectors a⃗=i^−j^+3k^\vec{a} = \hat{i} - \hat{j} + 3\hat{k} and b⃗=2i^−7j^+k^\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}.
  11. Ex 10.4 Q11
    Let the vectors a⃗\vec{a} and b⃗\vec{b} be such that ∣a⃗∣=3|\vec{a}| = 3 and ∣b⃗∣=23|\vec{b}| = \frac{\sqrt{2}}{3}, then a⃗×b⃗\vec{a} \times \vec{b} is a unit vector, if the angle between a⃗\vec{a} and b⃗\vec{b} is

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  12. Ex 10.4 Q12
    Area of a rectangle having vertices A, B, C and D with position vectors −i^+12j^+4k^-\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k}, i^+12j^+4k^\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k}, i^−12j^+4k^\hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} and −i^−12j^+4k^-\hat{i} - \frac{1}{2}\hat{j} + 4\hat{k}, respectively is

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Miscellaneous Exercise on Chapter 10

24 q

Solved Examples

Worked · 5
  1. Misc Eg.26
    Write all the unit vectors in XY-plane.
  2. Misc Eg.27
    If i^+j^+k^\hat{i} + \hat{j} + \hat{k}, 2i^+5j^2\hat{i} + 5\hat{j}, 3i^+2j^−3k^3\hat{i} + 2\hat{j} - 3\hat{k} and i^−6j^−k^\hat{i} - 6\hat{j} - \hat{k} are the position vectors of points A, B, C and D respectively, then find the angle between AB→\overrightarrow{\mathrm{AB}} and CD→\overrightarrow{\mathrm{CD}}. Deduce that AB→\overrightarrow{\mathrm{AB}} and CD→\overrightarrow{\mathrm{CD}} are collinear.
  3. Misc Eg.28
    Let a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} be three vectors such that ∣a⃗∣=3|\vec{a}| = 3, ∣b⃗∣=4|\vec{b}| = 4, ∣c⃗∣=5|\vec{c}| = 5 and each one of them being perpendicular to the sum of the other two, find ∣a⃗+b⃗+c⃗∣|\vec{a} + \vec{b} + \vec{c}|.
  4. Misc Eg.29
    Three vectors a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} satisfy the condition a⃗+b⃗+c⃗=0⃗\vec{a} + \vec{b} + \vec{c} = \vec{0}. Evaluate the quantity μ=a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗\mu = \vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}, if ∣a⃗∣=3|\vec{a}| = 3, ∣b⃗∣=4|\vec{b}| = 4 and ∣c⃗∣=2|\vec{c}| = 2.
  5. Misc Eg.30
    If with reference to the right handed system of mutually perpendicular unit vectors i^,j^\hat{i}, \hat{j} and k^\hat{k}, α⃗=3i^−j^\vec{\alpha} = 3\hat{i} - \hat{j}, β⃗=2i^+j^−3k^\vec{\beta} = 2\hat{i} + \hat{j} - 3\hat{k}, then express β⃗\vec{\beta} in the form β⃗=β1⃗+β2⃗\vec{\beta} = \vec{\beta_{1}} + \vec{\beta_{2}}, where β1⃗\vec{\beta_{1}} is parallel to α⃗\vec{\alpha} and β2⃗\vec{\beta_{2}} is perpendicular to α⃗\vec{\alpha}.

Miscellaneous Exercise

Practice · 19
  1. Misc Q1
    Write down a unit vector in XY-plane, making an angle of 30∘30^{\circ} with the positive direction of xx-axis.
  2. Misc Q2
    Find the scalar components and magnitude of the vector joining the points P(x1,y1,z1)\mathrm{P}(x_{1}, y_{1}, z_{1}) and Q(x2,y2,z2)\mathrm{Q}(x_{2}, y_{2}, z_{2}).
  3. Misc Q3
    A girl walks 4 km towards west, then she walks 3 km in a direction 30∘30^{\circ} east of north and stops. Determine the girl's displacement from her initial point of departure.
  4. Misc Q4
    If a⃗=b⃗+c⃗\vec{a} = \vec{b} + \vec{c}, then is it true that ∣a⃗∣=∣b⃗∣+∣c⃗∣|\vec{a}| = |\vec{b}| + |\vec{c}|? Justify your answer.
  5. Misc Q5
    Find the value of xx for which x(i^+j^+k^)x(\hat{i} + \hat{j} + \hat{k}) is a unit vector.
  6. Misc Q6
    Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a⃗=2i^+3j^−k^\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k} and b⃗=i^−2j^+k^\vec{b} = \hat{i} - 2\hat{j} + \hat{k}.
  7. Misc Q7
    If a⃗=i^+j^+k^\vec{a} = \hat{i} + \hat{j} + \hat{k}, b⃗=2i^−j^+3k^\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} and c⃗=i^−2j^+k^\vec{c} = \hat{i} - 2\hat{j} + \hat{k}, find a unit vector parallel to the vector 2a⃗−b⃗+3c⃗2\vec{a} - \vec{b} + 3\vec{c}.
  8. Misc Q8
    Show that the points A(1,−2,−8)\mathrm{A}(1, -2, -8), B(5,0,−2)\mathrm{B}(5, 0, -2) and C(11,3,7)\mathrm{C}(11, 3, 7) are collinear, and find the ratio in which B divides AC.
  9. Misc Q9
    Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a⃗+b⃗)(2\vec{a} + \vec{b}) and (a⃗−3b⃗)(\vec{a} - 3\vec{b}) externally in the ratio 1:21 : 2. Also, show that P is the mid point of the line segment RQ.
  10. Misc Q10
    The two adjacent sides of a parallelogram are 2i^−4j^+5k^2\hat{i} - 4\hat{j} + 5\hat{k} and i^−2j^−3k^\hat{i} - 2\hat{j} - 3\hat{k}. Find the unit vector parallel to its diagonal. Also, find its area.
  11. Misc Q11
    Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ±(13,13,13)\pm\left(\dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}\right).
  12. Misc Q12
    Let a⃗=i^+4j^+2k^\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}, b⃗=3i^−2j^+7k^\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k} and c⃗=2i^−j^+4k^\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}. Find a vector d⃗\vec{d} which is perpendicular to both a⃗\vec{a} and b⃗\vec{b}, and c⃗⋅d⃗=15\vec{c}\cdot\vec{d} = 15.
  13. Misc Q13
    The scalar product of the vector i^+j^+k^\hat{i} + \hat{j} + \hat{k} with a unit vector along the sum of vectors 2i^+4j^−5k^2\hat{i} + 4\hat{j} - 5\hat{k} and λi^+2j^+3k^\lambda\hat{i} + 2\hat{j} + 3\hat{k} is equal to one. Find the value of λ\lambda.
  14. Misc Q14
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are mutually perpendicular vectors of equal magnitudes, show that the vector c⃗⋅d⃗=15\vec{c}\cdot\vec{d} = 15 is equally inclined to a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c}.
  15. Misc Q15
    Prove that (a⃗+b⃗)⋅(a⃗+b⃗)=∣a⃗∣2+∣b⃗∣2(\vec{a} + \vec{b})\cdot(\vec{a} + \vec{b}) = |\vec{a}|^{2} + |\vec{b}|^{2}, if and only if a⃗,b⃗\vec{a}, \vec{b} are perpendicular, given a⃗≠0⃗,b⃗≠0⃗\vec{a} \ne \vec{0}, \vec{b} \ne \vec{0}.
  16. Misc Q16
    If θ\theta is the angle between two vectors a⃗\vec{a} and b⃗\vec{b}, then a⃗⋅b⃗≥0\vec{a}\cdot\vec{b} \ge 0 only when

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  17. Misc Q17
    Let a⃗\vec{a} and b⃗\vec{b} be two unit vectors and θ\theta is the angle between them. Then a⃗+b⃗\vec{a} + \vec{b} is a unit vector if

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  18. Misc Q18
    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}) is

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  19. Misc Q19
    If θ\theta is 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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