3D GEOMETRY
STRAIGHT LINES
1. Find the vector and the Cartesian equations of the median AD of the triangle ABC with
the vertices at A(2,3,4), B(1,6,9) and C(0,2,3)
2. Find the angle between

r  2 î  ĵ  k̂  μ (2 î  ĵ  2k̂ ) .
the
lines

r  î  2ĵ  k̂  λ (3 î  2 ĵ  2k̂ ) and
3. Find the foot of the perpendicular from the point
( 2,-1,5) in the line
x  11 y  2 z  8


10
 4  11
4. Find the image of the point ( 1,0,4) in the line
x2 y4 z


1
5 5
5. Find the point of intersection between the lines
x 1 y  3 z  5
and


3
5
7
x2 y4 z6
if they intersect.


1
3
5
6. Find the distance between the point (2,-1,5) and the line
x  11 y  2 z  8


10
 4  11
7. Find the shortest distance between the
lines whose vector equations are


r  (î  2ĵ  k̂)  λ (î  ĵ  k̂ ) and r  (2î  ĵ  k̂ )  μ (2 î  ĵ  2k̂ )
8. Find the shortest distance between the
lines whose equations are


r  (î  ĵ)  λ (2 î  ĵ  k̂ ) and r  (2î  ĵ  k̂)  μ (3 î  5 ĵ  2k̂ )
9. Find the shortest distance between lines whose equations are
and x 
y2 z3

2
3
x2 y 6 z 3


2
3
4
10.Find the shortest distance and the equation of line of line of shortest distance between
lines whose equations are
x 1 y  2 x 1
x  2 y 1 z 1
and




1
1
1
2
1
2
11.Find the shortest distance and the equation of line of line of shortest distance between
lines whose equations are
x 1 y  2 x 1
x  2 y 1 z 1
and




1
1
1
2
1
2
12.Find the shortest distance and the equation of line of line of shortest distance between

r  (3  t )î  (4  2t )ĵ  ( t  2)k̂
lines
whose
equations
are
and

r  (1  s )î  (3s - 7 )ĵ  (2s  2)k̂
Planes
13.Fid the vector equation of the plane in the dot product form, through the point (2,-1,5)
and perpendicular to the vector î  2ĵ  k̂
14.Fid the cartesian equation of the plane in the normal form, through the point (-2,-6,5)
and perpendicular to the vector 3 î  2ĵ  6k̂
15.Fid the cartesian equation of the plane in the normal form, through the point (-2,-6,5)
which is also the foot of the perpendicular from the point (2,3,2)
16.Fid the vector equation of the plane in the dot product form, through the points (2,-1,5),
(1,-3, 4) and (4,-2,0).
17. Fid the equation of the plane through the points (0,6,4), (1,5,4) and (4,-2,0).
18.Prove that the points (0,6,4), (1,5,4) and (2,4,4) and (2,6,2)are coplanar. Also find the
equation of the plane containing these points.
19. Fid the equation of the plane through the points (0,6,4), (1,5,4) and perpendicular to
the plane 3x+2y-3z=4
20.Fid the equation of the plane through the points (0,6,4), (1,5,4) and perpendicular to

the plane r  (3 î  2ĵ  6k̂)  3
21.Fid the equation of the plane through the points (1,-3,4) and perpendicular to the planes

r  (3î  2ĵ  6k̂)  3 and 3x-2y-+2z=1
22.A plane meets the x, y and z axes at A, B and C respectively such that the centroid of
the triangle ABC is (-3,2,4).
23.A plane meets the x, y and z axes at A, B and C respectively such that the centroid of
the triangle ABC is (p,q,r). Prove that the equation of the plane is
24.Find the point of intersection between the lines
x y z
  3
p q r
x 1 y z 1
and the plane 2x

1
2 2
y+2z=0.
25.Find the point of intersection between the line
x 1 y  2 z  3
and the plane


2
3
4

r  (î  ĵ  4k̂)  6
26.Find the angle between the line
x3 y2 z3


3
2
6
and
the
plane

r  (2î  ĵ  2k̂)  6

27.Find the angle between the line r  (3  t )î  (4  2t )ĵ  ( t  2)k̂ and the plane

r  (3î  2 ĵ  k̂)  6
28.Find the image of the point ( -1, 2, 0) in the plane x-2y+2z=1
29.Fid the equation of the plane through the points (0,6,4), (1,5,4) and parallel to the line

r  (1  2t )î  (3  2t )ĵ  (3t  2)k̂
30.Fid the equation of the plane through the point ( -1, -2, 0) and containing the line
x 1 y  2 z  3


2
3
2
31.Fid the equation of the plane through the point ( 2, -2, 1) and containing the line

r  (2 î  ĵ  k̂ )  λ (î  2 ĵ)
32.Fid the equation of the plane containing the lines
x2 y4 z6


1
3
5
x 1 y  3 z  5


and
3
5
7

33.Fid the equation of the plane containing the lines r  (2 î  ĵ  k̂ )  λ (4î  2 ĵ  2k̂)
and
x2 y4 z6


2
1
1
34.Fid the equation of the plane through the point (1,5,4) perpendicular to the plane

3x+2y-3z=4 and parallel to the line r  (3  2 s ) î  (s - 7 )ĵ  (2 s  2)k̂
35.Find the distance between the point (2,-1,5) and the plane x+2y-2z=4

36.Find the distance between the planes x+2y-2z=4 and r  (2î  4 ĵ  4k̂)  6
37.Find the distance of the point (2,-1,1) from the plane x+2y-2z=4 measuring parallel to
the line
x  2 y 1 z 1


2
1
2
38.Find the distance of the point (2,0,1) from the plane x+2y-2z=4 measuring parallel to
the line
x 1 y z 1
 
1
1 2
39.Find the distance of the point (2,-1,5) from the plane x+2y-2z=4 measuring parallel to
the line
x  2 y 1 z 1


2
1
2
40.Find the distance of the point (2,-1,5) from the line
x  2 y 1 z 1
measuring


2
1
2

along the plane r  ( 2î  ĵ  2k̂)  1  0
41.Fid the equation of the plane through the point (1,-2,0) and through the point of
intersection of the planes to the planes 3x+2y-z=4 and x+y-2z-2=0.
42.Fid the equation of the plane through the point of intersection of the planes to the
planes x+2y+z=4 and 3x+y-z-1=0 and through the origin.
43.Fid the equation of the plane through the point of intersection of the planes to the
planes x+2y-z=1 and x+y+2z+1=0 perpendicular to the plane 2x+2y-z=0.
44.Find the equation of the planes bisecting the angle between the planes 2x+2y-z=1 and
3x+2y+6z+1=0. Also indicate the equation of the plane bisecting the acute angle
between them.
45.Find the equation of the planes bisecting the angle between the planes x+2y-2z =2 and
2x-6y+3z+1=0. Also indicate the equation of the plane bisecting the obtuse angle
between them.
46.Fid the vector equation of the following plane in the dot product form

r  (î  ĵ)  λ (2 î  ĵ  k̂ )  μ (3 î  5 ĵ  2k̂ )
47.Fid the vector equation of the following plane in the dot product form

r  (2î  ĵ  k̂)  λ ( î  ĵ)  μ (î  2 ĵ  2k̂ )
48.Find the equation of the line x+2y-2z -2=0= x-2y+3z+1in the symmetrical form.
49.Find the equation of the line 2x+y-z -2=0=2x+3y-5z+1in the symmetrical form.
50.A variable plane which remains at a constant distance 3p from the origin cuts the
coordinate axes at A, B and C. Prove that the locus of the centroid of the triangle ABC
is
1
1 1
1
 2 2  2
2
x
y
z
p