# RD SHARMA Solutions for Class 12-science Maths Chapter 27 - Direction Cosines and Direction Ratios

## Chapter 27 - Direction Cosines and Direction Ratios Exercise Ex. 27.1

If a line makes angles of 90°, 60° and 30° with the positive direction of x,y and z-axis respectively, find its direction cosines.

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

If a line has direction ratios 2, -1, -2, determine its direction cosines.

Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3).

Using direction ratios show that the points *A* (2, 3, -4), *B *(1, -2, 3) and *C *(3, 8, -11) 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)

Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2.

Find the angle between the lines whose direction cosines are given by equations

2l + 2m - n = 0, mn + ln + lm = 0

## Chapter 27 - Direction Cosines and Direction Ratios Exercise MCQ

For every point P(x, y, z) on the xy-plane,

- x = 0
- y = 0
- z = 0
- x = y = z = 0

Correct option: (c)

Every point on xy-plane z co-ordinate is always zero.

For every point P(x, y, z) on the x-axis (except the origin),

- x = 0, y = 0, z ≠ 0
- x = 0, z = 0, y ≠ 0
- y = 0, z = 0, x ≠ 0
- x = y = z = 0

Correct option: (c)

Point (x, y, z) is on the x-axis. Hence, y and z co-ordinate will be zero except the origin.

A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is

- 2
- 3
- 4
- all of these

Correct option: (d)

Coordinates of the points given are diagonally opposite vertices of a parallelepiped. Hence, edges of parallelepiped can be 5-2, 7-3, 9-7 ⇒3,4,2.

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of a diagonal of the parallelopiped is

Correct option : (a)

The xy-plane divides the line joining the points (-1, 3, 4) and (2, -5, 6)

- internally in the ratio 2 : 3
- externally in the ratio 2 : 3
- internally in the ratio 3 : 2
- externally in the ratio 3 : 2

Correct option: (b)

If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, -2) is 4, then its z-coordinate is

- 2
- 1
- -1
- -2

Correct option: (c)

The distance of the point P(a, b, c) from the x-axis is

Correct option: (a)

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

- 3 : 4 internally
- 3 : 1 externally
- 1 : 2 internally
- 2 : 1 externally

Correct option: (b)

If P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear, then R divides PQ in the ratio

- 3 : 2 internally
- 3 : 2 externally
- 2 : 1 internally
- 2 : 1 externally

Correct option: (b)

A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are the vertices of a triangle ABC. If the bisector of ∠BAC meets BC at D, then coordinates of D are

- (19/8, 57/16, 17/16)
- (-19/8, 57/16, 17/16)
- (19/8, -57/16, 17/16)
- none of these

Correct option: (a)

If O is the origin, OP = 3 with direction ratios proportional to -1, 2, -2 then the coordinates of P are

- (-1, 2, -2)
- (1, 2, 2)
- (-1/9, 2/9, -2/9)
- (3, 6, -9)

Correct option: (a)

Directions of OP from the origin

(-1,2,-2)=(x,y,z)

The angle between the two diagonals of a cube is

Correct option: (d)

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos^{2}α + cos^{2}β + cos^{2}γ + cos^{2}δ is equal to

Correct option: (c)

## Chapter 27 - Direction Cosines and Direction Ratios Exercise Ex. 27VSAQ

So the projection of the point *P* on *y*-axis is (0, -3, 0)

If a line has direction ratios proportional to 2, -1, -2, then what are its direction cosines?

Write direction cosines of a line parallel to -axis.

If a unit vector makes an angle and an acute angle θ with k, then find the value of θ.

Write the distance of a point *P* (a, b, c) from x-axis.

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