Three dimensional geometry

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.

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.
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.

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 .
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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—Earl Nightingale

Three dimensional geometry

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