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Mathematics · Textbook solutions

Line and Planes

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. Plus every board question asked from this chapter. · 134 questions

6.1 Vector and Cartesian Equations of a Line

22 q

Solved Examples

Worked · 11
  1. 6.1 SolvedEx.1
    Verify that point having position vector 4i^−11j^+2k^4\hat{i}-11\hat{j}+2\hat{k} lies on the line rˉ=(6i^−4j^+5k^)+λ(2i^+7j^+3k^)\bar{r}=(6\hat{i}-4\hat{j}+5\hat{k})+\lambda(2\hat{i}+7\hat{j}+3\hat{k}).
  2. 6.1 SolvedEx.2
    Find the vector equation of the line passing through the point having position vector 4i^−j^+2k^4\hat{i}-\hat{j}+2\hat{k} and parallel to vector −2i^−j^+k^-2\hat{i}-\hat{j}+\hat{k}.
  3. 6.1 SolvedEx.3
    Find the vector equation of the line passing through the point having position vector 2i^+j^−3k^2\hat{i}+\hat{j}-3\hat{k} and perpendicular to vectors i^+j^+k^\hat{i}+\hat{j}+\hat{k} and i^+2j^−k^\hat{i}+2\hat{j}-\hat{k}.
  4. 6.1 SolvedEx.4
    Find the vector equation of the line passing through 2i^+j^−k^2\hat{i}+\hat{j}-\hat{k} and parallel to the line joining points −i^+j^+4k^-\hat{i}+\hat{j}+4\hat{k} and i^+2j^+2k^\hat{i}+2\hat{j}+2\hat{k}.
  5. 6.1 SolvedEx.5
    Find the vector equation of the line passing through A(1,2,3)A(1,2,3) and B(2,3,4)B(2,3,4).
  6. 6.1 SolvedEx.6
    Find the Cartesian equations of the line passing through A(1,2,3)A(1,2,3) and having direction ratios 2,3,72,3,7.
  7. 6.1 SolvedEx.7
    Find the Cartesian equations of the line passing through A(1,2,3)A(1,2,3) and B(2,3,4)B(2,3,4).
  8. 6.1 SolvedEx.8
    Find the Cartesian equations of the line passing through the point A(2,1,−3)A(2,1,-3) and perpendicular to vectors bˉ=i^+j^+k^\bar{b}=\hat{i}+\hat{j}+\hat{k} and cˉ=i^+2j^−k^\bar{c}=\hat{i}+2\hat{j}-\hat{k}.
  9. 6.1 SolvedEx.9
    Find the angle between lines rˉ=(i^+2j^+3k^)+λ(2i^−2j^+k^)\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}-2\hat{j}+\hat{k}) and rˉ=(i^+2j^+3k^)+λ(i^+2j^+2k^)\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}+2\hat{j}+2\hat{k}).
  10. 6.1 SolvedEx.10
    Show that lines rˉ=(−i^−3j^+4k^)+λ(−10i^−j^+k^)\bar{r}=(-\hat{i}-3\hat{j}+4\hat{k})+\lambda(-10\hat{i}-\hat{j}+\hat{k}) and rˉ=(−10i^−j^+k^)+μ(−i^−3j^+4k^)\bar{r}=(-10\hat{i}-\hat{j}+\hat{k})+\mu(-\hat{i}-3\hat{j}+4\hat{k}) intersect each other. Find the position vector of their point of intersection.
  11. 6.1 SolvedEx.11
    Find the co-ordinates of points on the line x+12=y−23=z+36\frac{x+1}{2}=\frac{y-2}{3}=\frac{z+3}{6}, which are at 3 unit distance from the base point A(−1,2,−3)A(-1,2,-3).

Exercise 6.1

Practice · 11
  1. Ex 6.1 Q.1
    Find the vector equation of the line passing through the point having position vector −2i^+j^+k^-2\hat{i}+\hat{j}+\hat{k} and parallel to vector 4i^−j^+2k^4\hat{i}-\hat{j}+2\hat{k}.
  2. Ex 6.1 Q.2
    Find the vector equation of the line passing through points having position vectors 3i^+4j^−7k^3\hat{i}+4\hat{j}-7\hat{k} and 6i^−j^+k^6\hat{i}-\hat{j}+\hat{k}.
  3. Ex 6.1 Q.3
    Find the vector equation of line passing through the point having position vector 5i^+4j^+3k^5\hat{i}+4\hat{j}+3\hat{k} and having direction ratios −3,4,2-3,4,2.
  4. Ex 6.1 Q.4
    Find the vector equation of the line passing through the point having position vector i^+2j^+3k^\hat{i}+2\hat{j}+3\hat{k} and perpendicular to vectors i^+j^+k^\hat{i}+\hat{j}+\hat{k} and 2i^−j^+k^2\hat{i}-\hat{j}+\hat{k}.
  5. Ex 6.1 Q.5
    Find the vector equation of the line passing through the point having position vector −i^−j^+2k^-\hat{i}-\hat{j}+2\hat{k} and parallel to the line rˉ=(i^+2j^+3k^)+λ(3i^+2j^+k^)\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(3\hat{i}+2\hat{j}+\hat{k}).
  6. Ex 6.1 Q.6
    Find the Cartesian equations of the line passing through A(−1,2,1)A(-1,2,1) and having direction ratios 2,3,12,3,1.
  7. Ex 6.1 Q.7
    Find the Cartesian equations of the line passing through A(2,2,1)A(2,2,1) and B(1,3,0)B(1,3,0).
  8. Ex 6.1 Q.8
    A(−2,3,4)A(-2,3,4), B(1,1,2)B(1,1,2) and C(4,−1,0)C(4,-1,0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
  9. Ex 6.1 Q.9
    Show that lines x+1−10=y+3−1=z−41\frac{x+1}{-10}=\frac{y+3}{-1}=\frac{z-4}{1} and x+10−1=y+1−3=z−14\frac{x+10}{-1}=\frac{y+1}{-3}=\frac{z-1}{4} intersect each other. Find the co-ordinates of their point of intersection.
  10. Ex 6.1 Q.10
    A line passes through (3,−1,2)(3,-1,2) and is perpendicular to lines rˉ=(i^+j^−k^)+λ(2i^−2j^+k^)\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2\hat{i}-2\hat{j}+\hat{k}) and rˉ=(2i^+j^−3k^)+μ(i^−2j^+2k^)\bar{r}=(2\hat{i}+\hat{j}-3\hat{k})+\mu(\hat{i}-2\hat{j}+2\hat{k}). Find its equation.
  11. Ex 6.1 Q.11
    Show that the line x−21=y−42=z+4−2\frac{x-2}{1}=\frac{y-4}{2}=\frac{z+4}{-2} passes through the origin.

6.2 Distance of a Point from a Line

2 q

Solved Examples

Worked · 2
  1. 6.2 SolvedEx.12
    Find the length of the perpendicular drawn from the point P(3,2,1)P(3,2,1) to the line rˉ=(7i^+7j^+6k^)+λ(−2i^+2j^+3k^)\bar{r}=\left(7\hat{i}+7\hat{j}+6\hat{k}\right)+\lambda\left(-2\hat{i}+2\hat{j}+3\hat{k}\right).
  2. 6.2 SolvedEx.13
    Find the distance of the point P(0,2,3)P(0,2,3) from the line x+35=y−12=z+43\dfrac{x+3}{5}=\dfrac{y-1}{2}=\dfrac{z+4}{3}.

6.3 Skew Lines and Distance Between Them

14 q

Solved Examples

Worked · 5
  1. 6.3 SolvedEx.14
    Find the shortest distance between lines rˉ=(2i^−j^)+λ(2i^+j^−3k^)\bar{r}=(2\hat{i}-\hat{j})+\lambda(2\hat{i}+\hat{j}-3\hat{k}) and rˉ=(i^−j^+2k^)+μ(2i^+j^−5k^)\bar{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(2\hat{i}+\hat{j}-5\hat{k}).
  2. 6.3 SolvedEx.15
    Find the shortest distance between lines x−12=y−23=z−34\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} and x−23=y−44=z−55\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}.
  3. 6.3 SolvedEx.16
    Show that lines rˉ=(i^+j^−k^)+λ(2i^−2j^+k^)\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2\hat{i}-2\hat{j}+\hat{k}) and rˉ=(4i^−3j^+2k^)+μ(i^−2j^+2k^)\bar{r}=(4\hat{i}-3\hat{j}+2\hat{k})+\mu(\hat{i}-2\hat{j}+2\hat{k}) intersect each other.
  4. 6.3 SolvedEx.17
    Find the distance between parallel lines rˉ=(2i^−j^+k^)+λ(2i^+j^−2k^)\bar{r}=(2\hat{i}-\hat{j}+\hat{k})+\lambda(2\hat{i}+\hat{j}-2\hat{k}) and rˉ=(i^−j^+2k^)+μ(2i^+j^−2k^)\bar{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(2\hat{i}+\hat{j}-2\hat{k}).
  5. 6.3 SolvedEx.18
    Find the distance between parallel lines x2=y−1=z2\dfrac{x}{2}=\dfrac{y}{-1}=\dfrac{z}{2} and x−12=y−1−1=z−12\dfrac{x-1}{2}=\dfrac{y-1}{-1}=\dfrac{z-1}{2}.

Exercise 6.2

Practice · 9
  1. Ex 6.2 Q.1
    Find the length of the perpendicular from (2,−3,1)(2,-3,1) to the line x+12=y−33=z+1−1\dfrac{x+1}{2}=\dfrac{y-3}{3}=\dfrac{z+1}{-1}.
  2. Ex 6.2 Q.2
    Find the co-ordinates of the foot of the perpendicular drawn from the point 2i^−j^+5k^2\hat{i}-\hat{j}+5\hat{k} to the line rˉ=(11i^−2j^−8k^)+λ(10i^−4j^−11k^)\bar{r}=\left(11\hat{i}-2\hat{j}-8\hat{k}\right)+\lambda\left(10\hat{i}-4\hat{j}-11\hat{k}\right). Also find the length of the perpendicular.
  3. Ex 6.2 Q.3
    Find the shortest distance between the lines rˉ=(4i^−j^)+λ(i^+2j^−3k^)\bar{r}=\left(4\hat{i}-\hat{j}\right)+\lambda\left(\hat{i}+2\hat{j}-3\hat{k}\right) and rˉ=(i^−j^+2k^)+μ(i^+4j^−5k^)\bar{r}=\left(\hat{i}-\hat{j}+2\hat{k}\right)+\mu\left(\hat{i}+4\hat{j}-5\hat{k}\right).
  4. Ex 6.2 Q.4
    Find the shortest distance between the lines x+17=y+1−6=z+11\dfrac{x+1}{7}=\dfrac{y+1}{-6}=\dfrac{z+1}{1} and x−31=y−5−2=z−71\dfrac{x-3}{1}=\dfrac{y-5}{-2}=\dfrac{z-7}{1}.
  5. Ex 6.2 Q.5
    Find the perpendicular distance of the point (1,0,0)(1,0,0) from the line x−12=y+1−3=z+108\dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+10}{8}. Also find the co-ordinates of the foot of the perpendicular.
  6. Ex 6.2 Q.6
    A(1,0,4)A(1,0,4), B(0,−11,13)B(0,-11,13), C(2,−3,1)C(2,-3,1) are three points and DD is the foot of the perpendicular from AA to BCBC. Find the co-ordinates of DD.
  7. By computing the shortest distance, determine whether following lines intersect each other.
    Ex 6.2 Q.7 i)
    rˉ=(i^−j^)+λ(2i^+k^)\bar{r}=\left(\hat{i}-\hat{j}\right)+\lambda\left(2\hat{i}+\hat{k}\right) and rˉ=(2i^−j^)+μ(i^+j^−k^)\bar{r}=\left(2\hat{i}-\hat{j}\right)+\mu\left(\hat{i}+\hat{j}-\hat{k}\right)
  8. Ex 6.2 Q.7 ii)
    x−54=y−7−5=z+3−5\dfrac{x-5}{4}=\dfrac{y-7}{-5}=\dfrac{z+3}{-5} and x−87=y−71=z−53\dfrac{x-8}{7}=\dfrac{y-7}{1}=\dfrac{z-5}{3}
  9. Ex 6.2 Q.8
    If lines x−12=y+13=z−14\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-1}{4} and x−31=y−k2=z1\dfrac{x-3}{1}=\dfrac{y-k}{2}=\dfrac{z}{1} intersect each other then find kk.

Miscellaneous Exercise 6 (A)

23 q
  1. Misc A Q.1
    Find the vector equation of the line passing through the point having position vector 3i^+4j^−7k^3\hat{i}+4\hat{j}-7\hat{k} and parallel to 6i^−j^+k^6\hat{i}-\hat{j}+\hat{k}.
  2. Misc A Q.2
    Find the vector equation of the line which passes through the point (3,2,1)(3, 2, 1) and is parallel to the vector 2i^+2j^−3k^2\hat{i}+2\hat{j}-3\hat{k}.
  3. Misc A Q.3
    Find the Cartesian equations of the line which passes through the point (−2,4,−5)(-2, 4, -5) and parallel to the line x+23=y−35=z+56\frac{x+2}{3}=\frac{y-3}{5}=\frac{z+5}{6}.
  4. Misc A Q.4
    Obtain the vector equation of the line x+53=y+45=z+56\frac{x+5}{3}=\frac{y+4}{5}=\frac{z+5}{6}.
  5. Misc A Q.5
    Find the vector equation of the line which passes through the origin and the point (5,−2,3)(5, -2, 3).
  6. Misc A Q.6
    Find the Cartesian equations of the line which passes through points (3,−2,−5)(3, -2, -5) and (3,−2,6)(3, -2, 6).
  7. Misc A Q.7
    Find the Cartesian equations of the line passing through A(3,2,1)A(3, 2, 1) and B(1,3,1)B(1, 3, 1).
  8. Misc A Q.8
    Find the Cartesian equations of the line passing through the point A(1,1,2)A(1, 1, 2) and perpendicular to vectors bˉ=i^+2j^+k^\bar{b}=\hat{i}+2\hat{j}+\hat{k} and cˉ=3i^+2j^−k^\bar{c}=3\hat{i}+2\hat{j}-\hat{k}.
  9. Misc A Q.9
    Find the Cartesian equations of the line which passes through the point (2,1,3)(2, 1, 3) and perpendicular to lines x−11=y−22=z−33\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} and x−3=y2=z5\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}.
  10. Misc A Q.10
    Find the vector equation of the line which passes through the origin and intersect the line x−1=y−2=z−3x-1=y-2=z-3 at right angle.
  11. Misc A Q.11
    Find the value of λ\lambda so that lines 1−x3=7y−142λ=z−32\frac{1-x}{3}=\frac{7y-14}{2\lambda}=\frac{z-3}{2} and 7−7x3λ=y−51=6−z5\frac{7-7x}{3\lambda}=\frac{y-5}{1}=\frac{6-z}{5} are at right angle.
  12. Misc A Q.12
    Find the acute angle between lines x−11=y−2−1=z−32\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-3}{2} and x−12=y−21=z−31\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{1}.
  13. Misc A Q.13
    Find the acute angle between lines x=y,z=0x=y, z=0 and x=0,z=0x=0, z=0.
  14. Misc A Q.14
    Find the acute angle between lines x=−y,z=0x=-y, z=0 and x=0,z=0x=0, z=0.
  15. Misc A Q.15
    Find the co-ordinates of the foot of the perpendicular drawn from the point (0,2,3)(0, 2, 3) to the line x+35=y−12=z+43\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}.
  16. By computing the shortest distance determine whether following lines intersect each other.
    Misc A Q.16 i)
    rˉ=(i^+j^−k^)+λ(2i^−j^+k^)\bar{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda\left(2\hat{i}-\hat{j}+\hat{k}\right) and rˉ=(2i^+2j^−3k^)+μ(i^+j^−2k^)\bar{r}=\left(2\hat{i}+2\hat{j}-3\hat{k}\right)+\mu\left(\hat{i}+\hat{j}-2\hat{k}\right).
  17. Misc A Q.16 ii)
    x−54=y−75=z+35\frac{x-5}{4}=\frac{y-7}{5}=\frac{z+3}{5} and x−6=y−8=z+2x-6=y-8=z+2.
  18. Misc A Q.17
    If lines x−12=y+13=z−14\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−21=y+m2=z−21\frac{x-2}{1}=\frac{y+m}{2}=\frac{z-2}{1} intersect each other then find mm.
  19. Misc A Q.18
    Find the vector and Cartesian equations of the line passing through the point (−1,−1,2)(-1, -1, 2) and parallel to the line 2x−2=3y+1=6z−22x-2=3y+1=6z-2.
  20. Misc A Q.19
    Find the direction cosines of the line rˉ=(−2i^+52j^−k^)+λ(2i^+3j^)\bar{r}=\left(-2\hat{i}+\frac{5}{2}\hat{j}-\hat{k}\right)+\lambda\left(2\hat{i}+3\hat{j}\right).
  21. Misc A Q.20
    Find the Cartesian equation of the line passing through the origin which is perpendicular to x−1=y−2=z−1x-1=y-2=z-1 and intersects the line x−12=y+13=z−14\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}.
  22. Misc A Q.21
    Write the vector equation of the line whose Cartesian equations are y=2y=2 and 4x−3z+5=04x-3z+5=0.
  23. Misc A Q.22
    Find the co-ordinates of points on the line x−11=y−2−2=z−32\frac{x-1}{1}=\frac{y-2}{-2}=\frac{z-3}{2} which are at the distance 3 unit from the base point A(1,2,3)A(1, 2, 3).

6.4 Equations of Plane

22 q

Solved Examples

Worked · 11
  1. 6.4 SolvedEx.1
    Find the vector equation of the plane passing through the point having position vector 2i^+3j^+4k^2\hat{i}+3\hat{j}+4\hat{k} and perpendicular to the vector 2i^+j^−2k^2\hat{i}+\hat{j}-2\hat{k}.
  2. 6.4 SolvedEx.2
    Find the Cartesian equation of the plane passing through A(1,2,3)A(1, 2, 3) and the direction ratios of whose normal are 3,2,53, 2, 5.
  3. 6.4 SolvedEx.3
    The foot of the perpendicular drawn from the origin to a plane is M(2,1,−2)M(2, 1, -2). Find the vector equation of the plane.
  4. 6.4 SolvedEx.4
    Find the vector equation of the plane passing through the point A(−1,2,−5)A(-1, 2, -5) and parallel to vectors 4i^−j^+3k^4\hat{i}-\hat{j}+3\hat{k} and i^+j^−k^\hat{i}+\hat{j}-\hat{k}.
  5. 6.4 SolvedEx.5
    Find the Cartesian equation of the plane rˉ=(i^−j^)+λ(i^+j^+k^)+μ(i^−2j^+3k^)\bar{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k}).
  6. 6.4 SolvedEx.6
    Find the vector equation of the plane passing through points A(1,1,2)A(1, 1, 2), B(0,2,3)B(0, 2, 3) and C(4,5,6)C(4, 5, 6).
  7. 6.4 SolvedEx.7
    Find the vector equation of the plane which is at a distance of 6 unit from the origin and to which the vector 2i^−j^+2k^2\hat{i}-\hat{j}+2\hat{k} is normal.
  8. 6.4 SolvedEx.8
    Find the perpendicular distance of the origin from the plane x−3y+4z−6=0x-3y+4z-6=0.
  9. 6.4 SolvedEx.9
    Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x+y−2z=182x+y-2z=18.
  10. 6.4 SolvedEx.10
    Reduce the equation rˉ⋅(3i^−4j^+12k^)=8\bar{r}\cdot(3\hat{i}-4\hat{j}+12\hat{k})=8 to the normal form and hence find (i) the length of the perpendicular from the origin to the plane (ii) direction cosines of the normal.
  11. 6.4 SolvedEx.11
    Find the vector equation of the plane passing through the point (1,0,2)(1, 0, 2) and the line of intersection of planes rˉ⋅(i^+j^+k^)=8\bar{r}\cdot(\hat{i}+\hat{j}+\hat{k})=8 and rˉ⋅(2i^+3j^+4k^)=3\bar{r}\cdot(2\hat{i}+3\hat{j}+4\hat{k})=3.

Exercise 6.3

Practice · 11
  1. Ex 6.3 Q.1
    Find the vector equation of a plane which is at 42 unit distance from the origin and which is normal to the vector 2i^+j^−2k^2\hat{i}+\hat{j}-2\hat{k}.
  2. Ex 6.3 Q.2
    Find the perpendicular distance of the origin from the plane 6x−2y+3z−7=06x-2y+3z-7=0.
  3. Ex 6.3 Q.3
    Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x+6y−3z=632x+6y-3z=63.
  4. Ex 6.3 Q.4
    Reduce the equation rˉ⋅(3i^+4j^+12k^)=78\bar{r}\cdot(3\hat{i}+4\hat{j}+12\hat{k})=78 to normal form and hence find (i) the length of the perpendicular from the origin to the plane (ii) direction cosines of the normal.
  5. Ex 6.3 Q.5
    Find the vector equation of the plane passing through the point having position vector i^+j^+k^\hat{i}+\hat{j}+\hat{k} and perpendicular to the vector 4i^+5j^+6k^4\hat{i}+5\hat{j}+6\hat{k}.
  6. Ex 6.3 Q.6
    Find the Cartesian equation of the plane passing through A(−1,2,3)A(-1, 2, 3), the direction ratios of whose normal are 0,2,50, 2, 5.
  7. Ex 6.3 Q.7
    Find the Cartesian equation of the plane passing through A(7,8,6)A(7, 8, 6) and parallel to the XY plane.
  8. Ex 6.3 Q.8
    The foot of the perpendicular drawn from the origin to a plane is M(1,0,0)M(1, 0, 0). Find the vector equation of the plane.
  9. Ex 6.3 Q.9
    Find the vector equation of the plane passing through the point A(−2,7,5)A(-2, 7, 5) and parallel to vectors 4i^−j^+3k^4\hat{i}-\hat{j}+3\hat{k} and i^+j^+k^\hat{i}+\hat{j}+\hat{k}.
  10. Ex 6.3 Q.10
    Find the Cartesian equation of the plane rˉ=(5i^−2j^−3k^)+λ(i^+j^+k^)+μ(i^−2j^+3k^)\bar{r}=(5\hat{i}-2\hat{j}-3\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k}).
  11. Ex 6.3 Q.11
    Find the vector equation of the plane which makes intercepts 1,1,11, 1, 1 on the co-ordinate axes.

6.5 Angle Between Planes

2 q

Solved Examples

Worked · 2
  1. 6.5 SolvedEx.12
    Find the angle between planes rˉ⋅(i^+j^−2k^)=8\bar{r}\cdot(\hat{i}+\hat{j}-2\hat{k})=8 and rˉ⋅(−2i^+j^+k^)=3\bar{r}\cdot(-2\hat{i}+\hat{j}+\hat{k})=3.
  2. 6.5 SolvedEx.13
    Find the angle between the line rˉ=(i^+2j^+k^)+λ(i^+j^+k^)\bar{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k}) and the plane rˉ⋅(2i^−j^+k^)=8\bar{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=8.

6.6 Coplanarity of Two Lines

1 q

Solved Examples

Worked · 1
  1. 6.6 SolvedEx.14
    Show that lines rˉ=(i^+j^−k^)+λ(2i^−2j^+k^)\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2\hat{i}-2\hat{j}+\hat{k}) and rˉ=(4i^−3j^+2k^)+μ(i^−2j^+2k^)\bar{r}=(4\hat{i}-3\hat{j}+2\hat{k})+\mu(\hat{i}-2\hat{j}+2\hat{k}) are coplanar. Find the equation of the plane determined by them.

6.7 Distance of a Point from a Plane

6 q

Solved Examples

Worked · 1
  1. 6.7 SolvedEx.15
    Find the distance of the point 4i^−3j^+2k^4\hat{i}-3\hat{j}+2\hat{k} from the plane rˉ⋅(−2i^+j^−2k^)=6\bar{r}\cdot(-2\hat{i}+\hat{j}-2\hat{k})=6.

Exercise 6.4

Practice · 5
  1. Ex 6.4 Q.1
    Find the angle between planes rˉ⋅(i^+j^+2k^)=13\bar{r}\cdot(\hat{i}+\hat{j}+2\hat{k})=13 and rˉ⋅(2i^−j^+k^)=31\bar{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=31.
  2. Ex 6.4 Q.2
    Find the acute angle between the line rˉ⋅(i^+2j^+2k^)+λ(2i^+3j^−6k^)\bar{r}\cdot(\hat{i}+2\hat{j}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}-6\hat{k}) and the plane rˉ⋅(2i^−j^+k^)=0\bar{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=0.
  3. Ex 6.4 Q.3
    Show that lines rˉ=(2j^−3k^)+λ(i^+2j^+3k^)\bar{r}=(2\hat{j}-3\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k}) and rˉ=(2i^+6j^+3k^)+μ(2i^+3j^+4k^)\bar{r}=(2\hat{i}+6\hat{j}+3\hat{k})+\mu(2\hat{i}+3\hat{j}+4\hat{k}) are coplanar. Find the equation of the plane determined by them.
  4. Ex 6.4 Q.4
    Find the distance of the point 4i^−3j^+k^4\hat{i}-3\hat{j}+\hat{k} from the plane rˉ⋅(2i^+3j^−6k^)=21\bar{r}\cdot(2\hat{i}+3\hat{j}-6\hat{k})=21.
  5. Ex 6.4 Q.5
    Find the distance of the point (1,1,−1)(1, 1, -1) from the plane 3x+4y−12z+20=03x+4y-12z+20=0.

Miscellaneous Exercise 6 (B)

42 q

Choose correct alternatives

Practice · 20
  1. Misc I Q.1
    If the line x3=y4=z\frac{x}{3}=\frac{y}{4}=z is perpendicular to the line x−1k=y+23=z−3k−1\frac{x-1}{k}=\frac{y+2}{3}=\frac{z-3}{k-1} then the value of kk is:
    1. A.
      114\frac{11}{4}
    2. B.
      −114-\frac{11}{4}
    3. C.
      112\frac{11}{2}
    4. D.
      411\frac{4}{11}
  2. Misc I Q.2
    The vector equation of line 2x−1=3y+2=z−22x-1=3y+2=z-2 is
    1. A.
      rˉ=(12i^−23j^+2k^)+λ(3i^+2j^+6k^)\bar{r}=\left(\frac{1}{2}\hat{i}-\frac{2}{3}\hat{j}+2\hat{k}\right)+\lambda\left(3\hat{i}+2\hat{j}+6\hat{k}\right)
    2. B.
      rˉ=i^−j^+(2i^+j^+k^)\bar{r}=\hat{i}-\hat{j}+\left(2\hat{i}+\hat{j}+\hat{k}\right)
    3. C.
      rˉ=(12i^−j^)+λ(i^−2j^+6k^)\bar{r}=\left(\frac{1}{2}\hat{i}-\hat{j}\right)+\lambda\left(\hat{i}-2\hat{j}+6\hat{k}\right)
    4. D.
      rˉ=(i^+j^)+λ(i^−2j^+6k^)\bar{r}=\left(\hat{i}+\hat{j}\right)+\lambda\left(\hat{i}-2\hat{j}+6\hat{k}\right)
  3. Misc I Q.3
    The direction ratios of the line which is perpendicular to the two lines x−72=y+17−3=z−61\frac{x-7}{2}=\frac{y+17}{-3}=\frac{z-6}{1} and x+51=y+32=z−6−2\frac{x+5}{1}=\frac{y+3}{2}=\frac{z-6}{-2} are
    1. A.
      4,5,74,5,7
    2. B.
      4,−5,74,-5,7
    3. C.
      4,−5,−74,-5,-7
    4. D.
      −4,5,8-4,5,8
  4. Misc I Q.4
    The length of the perpendicular from (1,6,3)(1,6,3) to the line x1=y−12=z−23\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}
    1. A.
      33
    2. B.
      11\sqrt{11}
    3. C.
      13\sqrt{13}
    4. D.
      55
  5. Misc I Q.5
    The shortest distance between the lines rˉ=(i^+2j^+k^)+λ(i^−j^−k^)\bar{r}=\left(\hat{i}+2\hat{j}+\hat{k}\right)+\lambda\left(\hat{i}-\hat{j}-\hat{k}\right) and rˉ=(2i^−j^−k^)+μ(2i^+j^+2k^)\bar{r}=\left(2\hat{i}-\hat{j}-\hat{k}\right)+\mu\left(2\hat{i}+\hat{j}+2\hat{k}\right) is
    1. A.
      13\frac{1}{\sqrt{3}}
    2. B.
      12\frac{1}{\sqrt{2}}
    3. C.
      32\frac{3}{\sqrt{2}}
    4. D.
      32\frac{\sqrt{3}}{2}
  6. Misc I Q.6
    The lines x−21=y−31=z−4−k\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k} and x−1k=y−42=z−51\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1} are coplanar if
    1. A.
      k=1k=1 or −1-1
    2. B.
      k=0k=0 or −3-3
    3. C.
      k=±3k=\pm 3
    4. D.
      k=0k=0 or −1-1
  7. Misc I Q.7
    The lines x1=y2=z3\frac{x}{1}=\frac{y}{2}=\frac{z}{3} and x−1−2=y−2−4=z−36\frac{x-1}{-2}=\frac{y-2}{-4}=\frac{z-3}{6} are
    1. A.
      perpendicular
    2. B.
      intersecting
    3. C.
      skew
    4. D.
      coincident
  8. Misc I Q.8
    Equation of X-axis is
    1. A.
      x=y=zx=y=z
    2. B.
      y=zy=z
    3. C.
      y=0, z=0y=0,\ z=0
    4. D.
      x=0, y=0x=0,\ y=0
  9. Misc I Q.9
    The angle between the lines 2x=3y=−z2x=3y=-z and 6x=−y=−4z6x=-y=-4z is
    1. A.
      45∘45^\circ
    2. B.
      30∘30^\circ
    3. C.
      0∘0^\circ
    4. D.
      90∘90^\circ
  10. Misc I Q.10
    The direction ratios of the line 3x+1=6y−2=1−z3x+1=6y-2=1-z are
    1. A.
      2,1,62,1,6
    2. B.
      2,1,−62,1,-6
    3. C.
      2,−1,62,-1,6
    4. D.
      −2,1,6-2,1,6
  11. Misc I Q.11
    The perpendicular distance of the plane 2x+3y−z=k2x+3y-z=k from the origin is 14\sqrt{14} units, the value of kk is
    1. A.
      1414
    2. B.
      196196
    3. C.
      2142\sqrt{14}
    4. D.
      142\frac{\sqrt{14}}{2}
  12. Misc I Q.12
    The angle between the planes rˉ⋅(i^−2j^+3k^)+4=0\bar{r}\cdot\left(\hat{i}-2\hat{j}+3\hat{k}\right)+4=0 and rˉ⋅(2i^+j^−3k^)+7=0\bar{r}\cdot\left(2\hat{i}+\hat{j}-3\hat{k}\right)+7=0 is
    1. A.
      π2\frac{\pi}{2}
    2. B.
      π3\frac{\pi}{3}
    3. C.
      cos⁡−1(34)\cos^{-1}\left(\frac{3}{4}\right)
    4. D.
      cos⁡−1(914)\cos^{-1}\left(\frac{9}{14}\right)
  13. Misc I Q.13
    If the planes rˉ⋅(2i^−λj^+k^)=3\bar{r}\cdot\left(2\hat{i}-\lambda\hat{j}+\hat{k}\right)=3 and rˉ⋅(4i^−j^+μk^)=5\bar{r}\cdot\left(4\hat{i}-\hat{j}+\mu\hat{k}\right)=5 are parallel, then the values of λ\lambda and μ\mu are respectively.
    1. A.
      12,−2\frac{1}{2},-2
    2. B.
      −12,2-\frac{1}{2},2
    3. C.
      −12,−2-\frac{1}{2},-2
    4. D.
      12,2\frac{1}{2},2
  14. Misc I Q.14
    The equation of the plane passing through (2,−1,3)(2,-1,3) and making equal intercepts on the coordinate axes is
    1. A.
      x+y+z=1x+y+z=1
    2. B.
      x+y+z=2x+y+z=2
    3. C.
      x+y+z=3x+y+z=3
    4. D.
      x+y+z=4x+y+z=4
  15. Misc I Q.15
    Measure of angle between the planes 5x−2y+3z−7=05x-2y+3z-7=0 and 15x−6y+9z+5=015x-6y+9z+5=0 is
    1. A.
      0∘0^\circ
    2. B.
      30∘30^\circ
    3. C.
      45∘45^\circ
    4. D.
      90∘90^\circ
  16. Misc I Q.16
    The direction cosines of the normal to the plane 2x−y+2z=32x-y+2z=3 are
    1. A.
      23,−13,23\frac{2}{3},\frac{-1}{3},\frac{2}{3}
    2. B.
      −23,13,−23\frac{-2}{3},\frac{1}{3},\frac{-2}{3}
    3. C.
      23,13,23\frac{2}{3},\frac{1}{3},\frac{2}{3}
    4. D.
      23,−13,−23\frac{2}{3},\frac{-1}{3},\frac{-2}{3}
  17. Misc I Q.17
    The equation of the plane passing through the points (1,−1,1)(1,-1,1), (3,2,4)(3,2,4) and parallel to Y-axis is:
    1. A.
      3x+2z−1=03x+2z-1=0
    2. B.
      3x−2z=13x-2z=1
    3. C.
      3x+2z+1=03x+2z+1=0
    4. D.
      3x+2z=23x+2z=2
  18. Misc I Q.18
    The equation of the plane in which the line x−54=y−74=z+3−5\frac{x-5}{4}=\frac{y-7}{4}=\frac{z+3}{-5} and x−87=y−41=z+53\frac{x-8}{7}=\frac{y-4}{1}=\frac{z+5}{3} lie, is
    1. A.
      17x−47y−24z+172=017x-47y-24z+172=0
    2. B.
      17x+47y−24z+172=017x+47y-24z+172=0
    3. C.
      17x+47y+24z+172=017x+47y+24z+172=0
    4. D.
      17x−47y+24z+172=017x-47y+24z+172=0
  19. Misc I Q.19
    If the line x+12=y−m3=z−46\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6} lies in the plane 3x−14y+6z+49=03x-14y+6z+49=0, then the value of mm is:
    1. A.
      55
    2. B.
      33
    3. C.
      22
    4. D.
      −5-5
  20. Misc I Q.20
    The foot of perpendicular drawn from the point (0,0,0)(0,0,0) to the plane is (4,−2,−5)(4,-2,-5) then the equation of the plane is
    1. A.
      4x+y+5z=144x+y+5z=14
    2. B.
      4x−2y−5z=454x-2y-5z=45
    3. C.
      x−2y−5z=10x-2y-5z=10
    4. D.
      4x+y+6z=114x+y+6z=11

Solve the following

Practice · 22
  1. Misc II Q.1
    Find the vector equation of the plane which is at a distance of 5 unit from the origin and which is normal to the vector 2i^+j^+2k^2\hat{i}+\hat{j}+2\hat{k}.
  2. Misc II Q.2
    Find the perpendicular distance of the origin from the plane 6x+2y+3z−7=06x+2y+3z-7=0.
  3. Misc II Q.3
    Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x+3y+6z=492x+3y+6z=49.
  4. Reduce the equation rˉ⋅(6i^+8j^+24k^)=13\bar{r}\cdot\left(6\hat{i}+8\hat{j}+24\hat{k}\right)=13 to normal form and hence find:
    Misc II Q.4 i)
    the length of the perpendicular from the origin to the plane.
  5. Misc II Q.4 ii)
    direction cosines of the normal.
  6. Misc II Q.5
    Find the vector equation of the plane passing through the points A(1,−2,1)A(1,-2,1), B(2,−1,−3)B(2,-1,-3) and C(0,1,5)C(0,1,5).
  7. Misc II Q.6
    Find the Cartesian equation of the plane passing through A(1,−2,3)A(1,-2,3) and the direction ratios of whose normal are 0,2,00,2,0.
  8. Misc II Q.7
    Find the Cartesian equation of the plane passing through A(7,8,6)A(7,8,6) and parallel to the plane rˉ⋅(6i^+8j^+7k^)=0\bar{r}\cdot\left(6\hat{i}+8\hat{j}+7\hat{k}\right)=0.
  9. Misc II Q.8
    The foot of the perpendicular drawn from the origin to a plane is M(1,2,0)M(1,2,0). Find the vector equation of the plane.
  10. Misc II Q.9
    A plane makes non zero intercepts a, b, ca,\ b,\ c on the co-ordinates axes. Show that the vector equation of the plane is rˉ⋅(bci^+caj^+abk^)=abc\bar{r}\cdot\left(bc\hat{i}+ca\hat{j}+ab\hat{k}\right)=abc.
  11. Misc II Q.10
    Find the vector equation of the plane passing through the point A(−2,3,5)A(-2,3,5) and parallel to vectors 4i^+3k^4\hat{i}+3\hat{k} and i^+j^\hat{i}+\hat{j}.
  12. Misc II Q.11
    Find the Cartesian equation of the plane rˉ=λ(i^+j^−k^)+μ(i^+2j^+3k^)\bar{r}=\lambda\left(\hat{i}+\hat{j}-\hat{k}\right)+\mu\left(\hat{i}+2\hat{j}+3\hat{k}\right).
  13. Misc II Q.12
    Find the vector equations of planes which pass through A(1,2,3)A(1,2,3), B(3,2,1)B(3,2,1) and make equal intercepts on the co-ordinates axes.
  14. Misc II Q.13
    Find the vector equation of the plane which makes equal non-zero intercepts on the co-ordinates axes and passes through (1,1,1)(1,1,1).
  15. Misc II Q.14
    Find the angle between planes rˉ⋅(−2i^+j^+2k^)=17\bar{r}\cdot\left(-2\hat{i}+\hat{j}+2\hat{k}\right)=17 and rˉ⋅(2i^+2j^+k^)=71\bar{r}\cdot\left(2\hat{i}+2\hat{j}+\hat{k}\right)=71.
  16. Misc II Q.15
    Find the acute angle between the line rˉ=λ(i^−j^+k^)\bar{r}=\lambda\left(\hat{i}-\hat{j}+\hat{k}\right) and the plane rˉ⋅(2i^−j^+k^)=23\bar{r}\cdot\left(2\hat{i}-\hat{j}+\hat{k}\right)=23.
  17. Misc II Q.16
    Show that lines rˉ=(i^+4j^)+λ(i^+2j^+3k^)\bar{r}=\left(\hat{i}+4\hat{j}\right)+\lambda\left(\hat{i}+2\hat{j}+3\hat{k}\right) and rˉ=(3j^−k^)+μ(2i^+3j^+4k^)\bar{r}=\left(3\hat{j}-\hat{k}\right)+\mu\left(2\hat{i}+3\hat{j}+4\hat{k}\right) are coplanar. Find the equation of the plane determined by them.
  18. Misc II Q.17
    Find the distance of the point 3i^+3j^+k^3\hat{i}+3\hat{j}+\hat{k} from the plane rˉ⋅(2i^+3j^+6k^)=21\bar{r}\cdot\left(2\hat{i}+3\hat{j}+6\hat{k}\right)=21.
  19. Misc II Q.18
    Find the distance of the point (13,13,−13)(13,13,-13) from the plane 3x+4y−12z=03x+4y-12z=0.
  20. Misc II Q.19
    Find the vector equation of the plane passing through the origin and containing the line rˉ=(i^+4j^+k^)+λ(i^+2j^+k^)\bar{r}=\left(\hat{i}+4\hat{j}+\hat{k}\right)+\lambda\left(\hat{i}+2\hat{j}+\hat{k}\right).
  21. Misc II Q.20
    Find the vector equation of the plane which bisects the segment joining A(2,3,6)A(2,3,6) and B(4,3,−2)B(4,3,-2) at right angle.
  22. Misc II Q.21
    Show that lines x=y, z=0x=y,\ z=0 and x+y=0, z=0x+y=0,\ z=0 intersect each other. Find the vector equation of the plane determined by them.