# 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 â€“ 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 90Â°, 135Â°, 45Â° 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 Î± = 90Â°, Î² = 135Â° and Î³ = 45Â°

So,

l = cos Î±, m = cos Î² and n = cos Î³

So direction cosines are

l = cos 90Â° = 0

m = cos 135Â°= cos (180Â° â€“ 45Â°) = -cos 45Â° = -1/âˆš2

n = cos 45Â° = 1/âˆš2

âˆ´ The direction cosines of the line are 0, -1/âˆš2, 1/âˆš2

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 Î±, Î², Î³

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

l = cos Î±, m = cos Î² and n = cos Î³

Here given Î± = Î² = Î³ (Since, line makes equal angles with the coordinate axes) â€¦ (1)

The direction cosines are

l = cos Î±, m = cos Î² and n = cos Î³

We have,

l2Â + mÂ 2Â + n2Â = 1

cos2Â Î± +Â cos2Î² + cos2Î³ = 1

From (1) we have,

cos2Â Î± + cos2Â Î± + cos2Â Î± = 1

3 cos2Â Î± = 1

Cos Î± = Â± âˆš(1/3)

âˆ´ The direction cosines are

l = Â± âˆš(1/3), m = Â± âˆš(1/3), n = Â± âˆš(1/3)

3. If a line has the direction ratios â€“18, 12, â€“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

âˆ´ The direction cosines are

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

4. Show that the points (2, 3, 4), (â€“1, â€“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(âˆ’1, âˆ’2, 1), C(5, 8, 7)

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

(âˆ’1âˆ’2), (âˆ’2âˆ’3), (1âˆ’4) = (âˆ’3, âˆ’5, âˆ’3)

Where, a1Â = -3, b1Â = -5, c1Â = -3

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

(5âˆ’ (âˆ’1)), (8âˆ’ (âˆ’2)), (7âˆ’1) = (6, 10, 6)

Where, a2Â = 6, b2Â = 10 and c2Â =6

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

By

âˆ´ A, B, C are collinear.

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

Solution:

Given:

The vertices are (3, 5, â€“4), (-1, 1, 2) and (â€“5, â€“5, â€“2).

The direction cosines of the two points passing through A(x1, y1, z1) and B(x2, y2, z2) is given by (x2Â â€“ 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 â€“ x1)2, (y2 â€“ y1)2, (z2 â€“ z1)2]

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

Then by using the formula,

âˆš[(x2 â€“ x1)2 + (y2 â€“ y1)2 + (z2 â€“ z1)2]

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

= âˆš68

= 2âˆš17

Now let us find the direction cosines of the line AB

By using the formula,

-4/2âˆš17 , -4/2âˆš17, 6/2âˆš17

Or -2/âˆš17, -2/âˆš17, 3/âˆš17

Similarly,

Let us find the direction ratios of BC

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

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

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

Then by using the formula,

âˆš[(x2 â€“ x1)2 + (y2 â€“ y1)2 + (z2 â€“ z1)2]

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

= âˆš68

= 2âˆš17

Now let us find the direction cosines of the line AB

By using the formula,

-4/2âˆš17, -6/2âˆš17, -4/2âˆš17

Or -2/âˆš17, -3/âˆš17, -2/âˆš17

Similarly,

Let us find the direction ratios of CA

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

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

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

Then by using the formula,

âˆš[(x2 â€“ x1)2 + (y2 â€“ y1)2 + (z2 â€“ z1)2]

âˆš[(8)2 + (10)2 + (-2)2] = âˆš(64+100+4)

= âˆš168

= 2âˆš42

Now let us find the direction cosines of the line AB

By using the formula,

8/2âˆš42, 10/2âˆš42, -2/2âˆš42

Or 4/âˆš42, 5/âˆš42, -1/âˆš42