 # NCERT Solutions for Class 12 Maths Exercise 11.1 Chapter 11, 3 Dimensional Geometry

## NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 – CBSE Term II Free PDF Download

The Exercise 11.1 of NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry is based on the following topics:

1. Introduction to Three Dimensional Geometry
2. Direction Cosines and Direction Ratios of a Line
1. The relation between the direction cosines of a line
2. Direction cosines of a line passing through two points

Solving the problems of this exercise of chapter 11 in Class 12 will help the students understand the topics mentioned above in a better way.

## Download PDF of NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry Exercise 11.1    ### Access Other Exercises of Class 12 Maths Chapter 11

Exercise 11.1 Solutions 5 Questions

Exercise 11.2 Solutions 17 Questions

Exercise 11.3 Solutions 14 Questions

Miscellaneous Exercise On Chapter 11 Solutions 23 Questions

#### Access Answers to NCERT Class 12 Maths Chapter 11 Exercise 11.1

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

Solution:

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

Here let α = 90°, β = 135° and γ = 45°

So,

l = cos α, m = cos β and n = cos γ

So direction cosines are

l = cos 90° = 0

m = cos 135°= cos (180° – 45°) = -cos 45° = -1/2

n = cos 45° = 1/2

∴ The direction cosines of the line are 0, -1/2, 1/2

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

Solution:

Given:

Angles are equal.

So let the angles be α, β, γ

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

l = cos α, m = cos β and n = cos γ

Here given α = β = γ (Since, line makes equal angles with the coordinate axes) … (1)

The direction cosines are

l = cos α, m = cos β and n = cos γ

We have,

l2 + m 2 + n2 = 1

cos2 α + cos2β + cos2γ = 1

From (1) we have,

cos2 α + cos2 α + cos2 α = 1

3 cos2 α = 1

Cos α = ± √(1/3)

∴ The direction cosines are

l = ± √(1/3), m = ± √(1/3), n = ± √(1/3)

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

Solution:

Given

Direction ratios as -18, 12, -4

Where, a = -18, b = 12, c = -4

Let us consider the direction ratios of the line as a, b and c

Then the direction cosines are ∴ The direction cosines are

-18/22, 12/22, -4/22 => -9/11, 6/11, -2/11

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

Solution:

If the direction ratios of two lines segments are proportional, then the lines are collinear.

Given:

A(2, 3, 4), B(−1, −2, 1), C(5, 8, 7)

Direction ratio of line joining A (2, 3, 4) and B (−1, −2, 1), are

(−1−2), (−2−3), (1−4) = (−3, −5, −3)

Where, a1 = -3, b1 = -5, c1 = -3

Direction ratio of line joining B (−1, −2, 1) and C (5, 8, 7) are

(5− (−1)), (8− (−2)), (7−1) = (6, 10, 6)

Where, a2 = 6, b2 = 10 and c2 =6

Hence it is clear that the direction ratios of AB and BC are of same proportions

By ∴ A, B, C are collinear.

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

Solution:

Given:

The vertices are (3, 5, –4), (-1, 1, 2) and (–5, –5, –2). The direction cosines of the two points passing through A(x1, y1, z1) and B(x2, y2, z2) is given by (x2 – x1), (y2-y1), (z2-z1)

Firstly let us find the direction ratios of AB

Where, A = (3, 5, -4) and B = (-1, 1, 2)

Ratio of AB = [(x2 – x1)2, (y2 – y1)2, (z2 – z1)2]

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

Then by using the formula,

√[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]

√[(-4)2 + (-4)2 + (6)2] = √(16+16+36)

= √68

= 2√17

Now let us find the direction cosines of the line AB

By using the formula, -4/2√17 , -4/2√17, 6/2√17

Or -2/√17, -2/√17, 3/√17

Similarly,

Let us find the direction ratios of BC

Where, B = (-1, 1, 2) and C = (-5, -5, -2)

Ratio of AB = [(x2 – x1)2, (y2 – y1)2, (z2 – z1)2]

= (-5+1), (-5-1), (-2-2) = -4, -6, -4

Then by using the formula,

√[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]

√[(-4)2 + (-6)2 + (-4)2] = √(16+36+16)

= √68

= 2√17

Now let us find the direction cosines of the line AB

By using the formula, -4/2√17, -6/2√17, -4/2√17

Or -2/√17, -3/√17, -2/√17

Similarly,

Let us find the direction ratios of CA

Where, C = (-5, -5, -2) and A = (3, 5, -4)

Ratio of AB = [(x2 – x1)2, (y2 – y1)2, (z2 – z1)2]

= (3+5), (5+5), (-4+2) = 8, 10, -2

Then by using the formula,

√[(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]

√[(8)2 + (10)2 + (-2)2] = √(64+100+4)

= √168

= 2√42

Now let us find the direction cosines of the line AB

By using the formula, 8/2√42, 10/2√42, -2/2√42

Or 4/√42, 5/√42, -1/√42