THREE DIMENSIONAL GEOMETRY
1.
9
6
2
, then what are its direction ratios?
11
1
2
3 2 6
− 3 , − 3 and 7 , 7 , 7. Find the angle between
If the direction cosines of a line are- 11 , 11 , −
2
2. The direction cosines of two lines are: 3 ,
them.
3. Find the angle between the lines whose direction ratios are < a, b, c > and < b – c , c – a , a – b >
4. Using direction numbers show that the points A(-2, 4, 7), B(3, -6, -8) and C(1, -2 , -2) are
collinear.
𝜋
5. A line makes an angle of 4 with each of X – axis and Y- axis. What angle does it make with Z –
axis.
6. A line in the XY- plane makes an angle of 30 with the x – axis. Find the direction cosines of the
line.
7. If a line makes angles  ,  ,  with the coordinate axes, prove that sin2 + sin2 +sin2 = 2.
8. Find the vector equation of the line passing through the point 2𝑖̂ - 𝐽̂ + 𝑘̂ and parallel to the
line joining the points −𝑖̂ +4 𝐽̂ + 5𝑘̂ and 𝑖̂ +2 𝐽̂ +2𝑘̂. Also find the Cartesian equation of
the line.
9. The Cartesian equation of a line are 2x -3 = 3y + 1 = 5 – 6z. Find the direction ratio of
the line and write down the vector equation of the line through (7, -5, 0) which is
parallel to the given line.
2𝑥−1
𝑦+2
𝑧−3
10. If the equation of a line AB is
= 2 = 3 . Find the direction cosines of a line
√3
11.
12.
13.
14.
15.
16.
17.
18.
parallel to AB.
Show the lines x = - y = 2z and x + 2 = 2y – 1 = - z + 1 are perpendicular to each other.
Find the equation of the plane passing through the point (2, 4, 6) and making equal
intercepts on axes.
Find the equation of the plane passing through the point (-1, 0, 7) and parallel to the
plane
3x – 5y + 4z = 11.
Find the distance between the planes 𝑟⃗(𝑖̂ + 2 𝐽̂ + 3𝑘̂ ) + 7 = 0 and 𝑟⃗( 2𝑖̂ + 4 𝐽̂ + 6𝑘̂ ) + 7 = 0.
Find the distance of the point (2, 5, -3) from the XY – plane.
Find the Cartesian and vector equations of the planes through the intersection of the
planes
2x + 6y + 12 = 0 and 3x – y + 4z = 0 which are at a unit distance from
the origin.
Find the equation of the plane passing through the line of intersection of the plane x –
2y + z =1 and 2x + y + z = 8 and parallel to the line with direction ratios <1 , 2, 1>. Also
find the perpendicular distance of P(1 , 3 2) from this plane.
A straight line passes through the point (2, -1, -1). It is parallel to the plane 4x + y +z +2
𝑥
𝑦
𝑧−5
= 0 and is perpendicular to the line 1 = −2 = 1 . Find its equation.
19. Show that the points (1, -1, 1), (2, 3, 1), (1 , 2 , 3) and (0, -2, 3) are coplanar. Also find the
equation of the plane containing them.
𝑥
20. Prove that the equation of the plane making intercepts a, b and c on the coordinate axes is +
𝑦
𝑏
𝑎
𝑧
+ 𝑐 = 1.
21. Find the equation of the plane which is parallel to x – axis and has intercepts 5 and 7 on y – axis
and z – axis respectively.
22. Find the distance of the point (1 , -2, 3) from the plane x – y + z = 5 measure along a line parallel
𝑥
𝑦
𝑧
to
= =
2
3
−6
23. Find the distance of the point A(-2, 3, -4) from the line
𝑥+2
3
=
2𝑦+3
4
=
3𝑧+ 4
−6
measured
parallel to the plane 4x + 12y – 3z + 1 = 0.
24. Find the equation of the line passing through the point (2, 1, 3) and perpendicular to the lines
𝑥−1
1
=
𝑦−2
2
=
𝑧−3
3
𝑥
and −3 =
𝑦
2
=
𝑧
.
5
25. Find the foot of the perpendicular drawn from the point P(1, 6, 3) on the line
Also find its distance from P.
𝑥
=
1
𝑦−1
2
=
𝑧−2
3
.