This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. It covers topics such as direction cosines, perpendicular lines and planes, intersections of lines and planes, and vector and Cartesian representations of geometric entities in three dimensions.

1. THREE DIMENSIONAL GEOMETRY
1 Mark Question :
1. If A(1, 5, 4) and B(4, 1, -2) find the direction ratio of 𝐴𝐵⃗⃗⃗⃗⃗ .
2. Write Direction ratios of line
𝑥−2
2
=
2𝑦−5
−3
= 𝑧 − 1 .
3. Write whether the line
𝑥−3
3
=
𝑦−2
1
=
𝑧−1
0
is perpendicular to x – axis, y-axis or z- axis.
4. Write the equation of plane passing through the points (a, 0,0) (0, b, 0) and (0, 0, c).
5. For what value of 𝛾the line
𝑥−1
2
=
𝑦−1
3
=
𝑧−1
𝛾
is perpendicular to normal plane 𝑟 =(2 𝑖̂+ 3𝑗̂ − 4𝑘̂) = 4 .
6. For what value of 𝛾, the planes x + 2y + 𝛾𝑧 = 18 and 2x – 4y + 3z = 7 are perpendicular to each other.
7. Write the direction cosines of line whose Cartesian equation are : 2x = 3y = -z .
8. If a line makes 𝛼, 𝛽, 𝛾 with x-axis , y-axis and z-axis respectively, then find the value of sin2
𝛼 + sin2
𝛽 + sin2
𝛾 .
9. Find the value of K for which the lines :
𝑥−1
−3
=
𝑦−2
2𝐾
=
𝑧−3
2
and
𝑥−1
3𝐾
=
𝑦−1
1
=
6−𝑧
5
are perpendicular to each other.
10. If the equation of the line 𝐴𝐵⃗⃗⃗⃗⃗ is
𝑥−3
1
=
𝑦+2
−2
=
𝑧−5
4
, find the direction ratios of a line parallel to AB .
4 Marks Questions :
11. Find the value of 𝜆 so that the lines
1−𝑥
3
=
7𝑦−14
2𝜆
=
5𝑧−10
11
and
7−7𝑥
3𝜆
=
𝑦−5
1
=
6−𝑧
5
are perpendicular to each other.
12. Find the distance of the point (-1, -5, -10) from the point of intersection of line 𝑟 = 2𝑖̂ − 𝑗̂ + 2𝑘̂ + 𝜆(3𝑖̂ + 4𝑗̂ + 2𝑘̂)
and the plane 𝑟 .(𝑖̂ − 𝑗̂ + 𝑘̂) = 5 .
13. Find the foot of perpendicular from P(1, 2, 3) on the line
𝑥−6
3
=
𝑦−7
2
=
𝑧−7
−2
. Also obtain the equation and length
of perpendicular.
14. Find the image of point (1, 6, 3) in the line
𝑥
1
=
𝑦−1
2
=
𝑧−2
3
15. Determine whether the following pairs of lines intersect or not :
𝑥−1
2
=
𝑦+1
3
= z,
𝑥+1
5
=
𝑦−2
1
=
𝑧−2
1
.
16. Find the shortest distance between the lines :
𝑥+1
7
=
𝑦+1
−6
=
𝑧+1
1
and
𝑥−3
1
=
𝑦−5
−2
=
𝑧−7
1
.
17. Find the shortest distance between the lines : 𝑟 = 𝑖̂ + 2𝑗̂ + 𝑘̂ + 𝜆( 𝑖̂ − 𝑗̂ + 𝑘̂); 𝑟 = 2𝑖̂ − 𝑗̂ − 𝑘̂ + 𝜇(2𝑖̂ + 𝑗̂ + 2𝑘̂) .
18. Show that the lines :
𝑥−1
2
=
𝑦−2
3
=
𝑧−3
4
and
𝑥−4
5
=
𝑦−1
2
= zintersect. Also find their point of intersection
19. Find whether the lines 𝑟 = ( 𝑖̂ − 𝑗̂ + 𝑘̂) + 𝜆(2𝑖̂ + 𝑗̂) and 𝑟 = (2𝑖̂ − 𝑗̂) + 𝜇( 𝑖̂ + 𝑗̂ − 𝑘̂) intersect or not. If
intersecting, find their point of intersection.
20. Find the coordinates of foot of perpendicular draw from the origin to the plane 2x – 3y + 4z = 6.
21. Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY –
plane.
22. Prove that if a plane has intercepts a, b, c and is at a distance of p units from origin, then
1
𝑎2
+
1
𝑏2
+
1
𝑐2
=
1
𝑝2
23. Find the length and foot of perpendicular from the point (1, 1, 2) to the plane 2x – 2y + 4z + 5 = 0.

2. 24. Show that the lines
𝑥−1
2
=
𝑦−3
4
=
𝑧
−1
and
𝑥−4
3
=
𝑦−1
−2
=
𝑧−1
1
are coplanar. Also find the equation of the plane containing these
lines.
25. Find the equation of plane passing through the points A(0, 0, 0) & B (3, -1, 2) and parallel to line
𝑥−4
1
=
𝑦+3
−4
=
𝑧+1
7
.
26. Find the equation of the plane passing through the point (-1, 2, 1) and perpendicular to the line joining
the points (-3, 1, 2) and (2, 3, 4). Also , find the perpendicular distance of the origin from this plane.
27. Find the perpendicular distance of the point (1, 0, 0) from the line
𝑥−1
2
=
𝑦+1
−3
=
𝑧+10
8
.
28. Find the angle between the line
𝑥+1
2
=
3𝑦+5
9
=
3−𝑧
−6
and the plane 10x + 2y – 11z = 3.
29. Find the equation of the perpendicular drawn from the point (1, -2, 3) to the plane 2x – 3y + 4z + 9 = 0 Also,
find the co-ordinates of the foot of the perpendicular.
30. Find the Vector and Cartesian equations of the line passing through the point (1, 2, -4) and perpendicular to the
two lines
𝑥−8
3
=
𝑦+19
−16
=
𝑧−10
7
and
𝑥−15
3
=
𝑦−29
8
=
𝑧−5
−5
.
6 Marks Questions :
1. Find the equation of the plane containing the line of intersection of the planes 2x – 3y + 5z = 7 and
3x +4y- z = 11 and passing through the point (1, 0, -2) .
2. Find the foot of perpendicular distance of the point (1, 3, 4) from the plane 2x – y + z + 3 = 0. Find also, the image
of the point in plane (-3, 5, 2) .
3. If the lines
𝑥−1
2
=
𝑦+1
3
=
𝑧−1
4
and
𝑥−3
1
=
𝑦−𝑘
2
=
𝑧
1
intersect, then find the value of k and hence find the equation of
plane containing these lines.
4. Find the equation of plane passing through the point (-1, 3, 2) and perpendicular to each of the planes
x + 2y + 3z = 5 and 3x + 3y + z = 0.
5. Find the equation of plane that contains the point (1, -1, 2) and is perpendicular to each of the planes 2x+3y-2z=5 &
x+2y-3z = 8 .
6. Find the vector equation of the plane passing through the intersection of planes 𝑟 . ( 𝑖̂ + 𝑗̂ + 𝑘̂) = 6 and
𝑟 .(2 𝑖̂+ 3𝑗̂ + 4𝑘̂) + 5 = 0 & point (1, 1, 1) .
7. Find the equation of plane passing through the points (3, 4, 1) & (0, 1, 0) & parallel to line
𝑥+3
2
=
𝑦−3
4
=
𝑧−2
5
.
8. Prove that the image of the point (3, -2, 1) in the plane 3x-y+4z = 2 lie on plane x+y+z+4 = 0.
9. Find the distance of points (-2, 3, -4) from the line
𝑥+2
3
=
2𝑦+3
4
=
3𝑧+4
5
measured parallel to plane
4x + 12y – 3z = 0.
10. Find the equation of plane through the line of intersection of planes x+y+z = 1 & 2x+3y+4z = 5. Which is
perpendicular to the plane x – y + z = 0.
11. Show that the lines
𝑥+3
−3
=
𝑦−1
1
=
𝑧−5
5
and
𝑥+1
−1
=
𝑦−2
2
=
𝑧−5
5
are coplanar. Also find the equation of plane
containing the lines.

3. 12. Find the equation of plane determined by the points A (3, -1, 2), B (5, 2, 4) & C (-1, -1, 6). Also find the distance
of point P(6,5, 9) from plane .[
6
√34
] .
13. Find the image of the point (1, 2, 3) in the plane x +2y + 4z = 38.
14. Show that the lines 𝑟 = ( 𝑖̂ + 𝑗̂ − 𝑘̂ ) + 𝛾(3𝑖̂ − 𝑗̂) and 𝑟 = (4𝑖̂ − 𝑘)̂ + 𝜇(2𝑖̂ + 3𝑘̂) are coplanar. Also find the plane
containing these two lines.
15. Find the equation of plane which contains two parallel lines
𝑥−3
3
=
𝑦+4
2
=
𝑧−1
1
and
𝑥+1
3
=
𝑦−2
2
=
𝑧
1
.
16. Find the equation of plane through the points (1, 2, 3) & (0, -1, 0) and parallel to the line
𝑟 .(2 𝑖̂+ 3𝑗̂ + 4𝑘̂) + 5 = 0 .
17. Find the Cartesian as well as vector equation of the planes , passing through the intersection of planes
𝑟 .(2 𝑖̂+ 6𝑗̂) + 12 = 0 &𝑟 .( 3𝑖̂ − 𝑗̂ + 4𝑘̂) = 0 , which are at a unit distance from origin.
18. Find the distance of the point (2, 2, -1) from the plane x + 2y – z = 1 measured parallel to the line
𝑥+1
1
=
𝑦+1
2
=
𝑧
3
.
19. Show that the points with position vectors 6𝑖̂ − 7𝑗̂ , 16𝑖̂ − 14𝑗̂ - 4𝑘̂ , 3𝑗̂ -6𝑘̂ and 2𝑖̂ − 5𝑗̂ + 10𝑘̂ are
coplanar .
20. Find the co-ordinates of the foot of perpendicular and the length of the perpendicular drawn from the point
P (5, 4, 2) to the line 𝑟 = - 𝑖̂ + 3𝑗̂ + 𝑘̂ + 𝜆 (2 𝑖̂ + 3𝑗̂ − 𝑘̂) . Also find the image of P in this line.
Find the image of the point having position vector 𝑖̂ + 3𝑗̂ + 4𝑘̂ in the planer 𝑟 (2 𝑖̂ − 𝑗̂ + 𝑘̂) +3 = 0 .
21. Show that the lines
𝑥+3
−3
=
𝑦−1
1
=
𝑧−5
5
,
𝑥+1
−1
=
𝑦−2
2
=
𝑧−5
5
are coplanar. Also find the equation of the plane
containing the lines.
22. Find the co-ordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane determined by
points A(1, 2, 3), B (2, 2, 1) and C (-1, 3, 6) .
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