Chapter-11. Three Dimensional Geometry

The planes: 2x - y + 4z = 5 and 5x - 2.5y + 10z = 6 are

(A) Perpendicular (B) Parallel (C) intersect y-axis

(C) passes through 

Distance between the two planes: and  is

(A)2 units (B)4 units (C)8 units

(D)

Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then 

Find the vector equation of the line passing through the point (1, 2, - 4) and perpendicular to the two lines: 

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .

Find the distance of the point ( - 1, - 5, - 10) from the point of intersection of the line and the plane.

Find the equation of the plane which contains the line of intersection of the planes ,  and which is perpendicular to the plane .

If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.

If the points (1, 1, p) and ( - 3, 0, 1) be equidistant from the plane , then find the value of p.

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Find the coordinates of the point where the line through (3, ­-4, -5) and (2, - 3, 1) crosses the plane 2x + y + z = 7).

Find the coordinates of the point where the line through (5, 1, 6) and

(3, 4, 1) crosses the YZ-plane

Find the shortest distance between lines 

and.

Find the equation of the plane passing through (a, b, c) and parallel to the plane 

Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane

If the lines and are perpendicular, find the value of k.

If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (­-4, 3, -6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

Find the equation of a line parallel to x-axis and passing through the origin.

Find the angle between the lines whose direction ratios are a, b, c and b - c,

c - a, a - b.

f l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂ - m₂n₁, n₁l₂ - n₂l₁, l₁m₂ ­- l₂m₁.

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1), (4, 3, -1).

In the following cases, find the distance of each of the given points from the corresponding given plane.

Point Plane

(a) (0, 0, 0) 

(b) (3, - 2, 1) 

(c) (2, 3, - 5) 

(d) ( - 6, 0, 0) 

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 

(b) 

(c) 

(d) 

(e) 

Find the angle between the planes whose vector equations are

and .

Find the equation of the plane through the line of intersection of the planes and which is perpendicular to the plane 

Find the vector equation of the plane passing through the intersection of the planes  and through the point (2, 1, 3)

Find the equation of the plane through the intersection of the planes  and  and the point (2, 2, 1)

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Find the intercepts cut off by the plane

Find the equations of the planes that passes through three points.

(a) (1, 1, -1), (6, 4, -5), (-4, -2, 3)

(b) (1, 1, 0), (1, 2, 1), (-2, 2, -1)

Find the vector and Cartesian equation of the planes

(a) that passes through the point (1, 0, - 2) and the normal to the plane is .

(b) that passes through the point (1, 4, 6) and the normal vector to the plane is .

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

(a)  (b) 

(c)  (d) 

Find the Cartesian equation of the following planes:

(a)  (b) 

(c) 

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

(a)z = 2 (b) 

(c)  (d)5y + 8 = 0

Find the shortest distance between the lines whose vector equations are

Find the shortest distance between the lines  and 

Find the shortest distance between the lines

Show that the lines and are perpendicular to each other.

Find the values of p so the line and

are at right angles.

Find the angle between the following pairs of lines:

(i) 

(ii) 

Find the angle between the following pairs of lines:

(i) 

(ii) and

Find the vector and the Cartesian equations of the line that passes through the points (3, -2, -5), (3, -2, 6).

Find the vector and the Cartesian equations of the lines that pass through the origin and (5, -2, 3).

The Cartesian equation of a line is . Write its vector form.

Find the Cartesian equation of the line which passes through the point

( - 2, 4, - 5) and parallel to the line given by

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector  and is in the direction .

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.

Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (-1, -2, 1), (1, 2, 5).

Show that the line through the points (1, -1, 2) (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Show that the three lines with direction cosines

 are mutually perpendicular.

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, - 4), (- 1, 1, 2) and (- 5, - 5, - 2)

Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.

If a line has the direction ratios -18, 12, -4, then what are its direction cosines?

Find the direction cosines of a line which makes equal angles with the coordinate axes.

If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.