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

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

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

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

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 three lines with direction cosines

 are mutually perpendicular.

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 line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through

the points (−1, −2, 1), (1, 2, 5).

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

Find the equation of the line in vector and in Cartesian form that passes through the point

with position vector 2i – j + 4k and is in the direction i + 2j – k.

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

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

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

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

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

Find the angle between the following pairs of lines:

(i) r = 2i – 5j + k + λ(3i – 2j + 6k) and r = 7i – 5k + μ(i + 2j + 2k)

(ii) r = 3i + j - 2k + λ(i – j - 2k) and r = 2i – j - 56k + μ(3i - 5j - 4k)

Find the angle between the following pairs of lines:

(i)

(ii)

Find the values of p so the line  and  are at right angles.

Show that the lines  are perpendicular to each other.

Find the shortest distance between the lines

r = i + 2j + k + λ(i – j + k) and r = 2i - j - k + μ(2i + j + 2k)

Find the shortest distance between the lines

 

Find the shortest distance between the lines whose vector equations are

r = i + 2j + 3k + λ(i – 3j + 2k) and r = 4i + 5j + 6k + μ(2i + 3j + k)

Find the shortest distance between the lines whose vector equations are

r = (1 - t)i + (t - 2)j + (3 – 2t)k and r = (s + 1)i + (2s - 1)j - (2s + 1)k

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) x + y + z = 1           

(c) 2x + 3y – z = 5               

(d) 5y + 8 = 0

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector 3i + 5j – 6k.

Find the Cartesian equation of the following planes:

(a) r.(i + j + k) = 2                             

(b) r.(2i + 3j – 4k) = 1       

(c) r.[(s – 2t)I + (3 - t)j + (2s + t)k] = 15

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

(a) 2x + 3y + 4z – 12 = 0                        

(b) 3y + 4x – 6 = 0       

(c) x + y + z = 1                                        

(d) 5y + 8 = 0

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 i + j - k.

(b) that passes through the point (1, 4, 6) and the normal vector to the plane is i – 2j + k.

Find the equation 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 intercepts cut off by the plane 2x + y – z = 5

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

Find the equation of the plane through the intersection of the planes

3x – y + 2z – 4 = 0 and x + y + z – 2 = 4 and the point (2, 2, 1).

Find the vector equation of the plane passing through the intersection of the planes

r.(2i + 2j – 3k) = 7, r.(2i + 5j + 3k) = 9 and through the point (2, 1, 3).

Find the equation of the plane through the line of intersection of the planes

x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Find the angle between the planes whose vector equations are                                                      

r.(2i + 2j – 3k) = 5 and r.(3i - 3j + 5k) = 3

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) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

(b) 2x + y + 3z - 2 = 0 and x – 2y + 5 = 0

(c) 2x - 2y + 4z + 5 = 0 and 3x – 3y + 6z - 1 = 0

(d) 2x - y + 3z - 1 = 0 and 2x – y + 3z + 3 = 0

(e) 4x + 8y + z - 8 = 0 and y + z - 4 = 0

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

      Point                                 Plane

 (a) (0, 0, 0)                    3x – 4y + 12z = 3

(b) (3, −2, 1)                   2x – y + 2z + 3 = 0

(c) (2, 3, −5)                    x + 2y – 2z = 9

(d) (−6, 0, 0)                   2x – 3y + 6z – 2 = 0

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).

If 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.

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

c − a, a − b.

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

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.

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

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

and perpendicular to the plane r.(i + 2j – 5k) + 9 = 0

Find the equation of the plane passing through (a, b, c) and parallel to the plane r.(i + j + k) = 2

Find the shortest distance between r = 6i + 2j + 2k + λ(i – 2j + 2k) and r = -4i – k + µ(3i – 2j – 2k).

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.

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 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.

If the points (1, 1, p) and (−3, 0, 1) be equidistant from the plane r.(3i + 4j – 12k) + 13 = 0, then find the value of p.

Find the equation of the plane passing through the line of intersection of the planes

r.(i + j + k) = 1 and r.(2i + 3k - k) + 4 = 0 and parallel to x-axis.

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.

Find the equation of the plane which contains the line of intersection of the planes

r.(i + 2j + 3k) – 4 = 0 and r.(2i + j - k) + 5 = 0 and which is perpendicular to the plane r.(5i + 3j – 6k) + 8 = 0

Find the distance of the point (−1, −5, −10) from the point of intersection of the line

r = 2i – j + 2k + λ(3i + 4j + 2k) and the plane r.(i – j + k) = 5

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes r.(i – j + 2k) = 5 and r.(3i + j + k) = 6

Find the vector equation of the line passing through the point (1, 2, − 4) and

perpendicular to the two lines:

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

Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is                                      

(A) 2 units                  

(B) 4 units                 

(C) 8 units                     

(D)  units

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

(A) Perpendicular                             

(B) Parallel         

(C) intersect y-axis                          

(D) passes through (0, 0, )