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

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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

NCERT Solutions for Class 12 Maths Chapter 11 –Three Dimensional Geometry image - 1

∴ 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

NCERT Solutions for Class 12 Maths Chapter 11 –Three Dimensional Geometry image - 2

∴ 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).

NCERT Solutions for Class 12 Maths Chapter 11 –Three Dimensional Geometry image - 3

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,

NCERT Solutions for Class 12 Maths Chapter 11 –Three Dimensional Geometry image - 4

-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,

NCERT Solutions for Class 12 Maths Chapter 11 –Three Dimensional Geometry image - 5

-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,

NCERT Solutions for Class 12 Maths Chapter 11 –Three Dimensional Geometry image - 6

8/2√42, 10/2√42, -2/2√42

Or 4/√42, 5/√42, -1/√42

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