# NCERT Solutions for Class 12 Maths Three Dimensional Geometry

Hi Students, Welcome to **Amans Maths Blogs (AMB)**. In this post, you will get the * NCERT Solutions for Class 12 Maths Three Dimensional Geometry Exercise 11.1*.

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**NCERT Solutions for Class 12 Maths 3D Geometry Exercise 11.1**The PDF books of * NCERT Solutions for Class 12* are the first step towards the learning and understanding the each sections of Maths, Physics, Chemistry, Biology as it all help in engineering medical entrance exams. To solve it, you just need to click on download links of

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**CBSE Class 12**^{th}Maths## NCERT Solutions for Class 12 Maths Three Dimensional Geometry

**NCERT Solutions for Class 12 Maths Three Dimensional Geometry ****Exercise**** 11.1 Ques No 1.**

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

**NCERT Solutions:**

If a line makes angles α, β and γ with x, y and z-axes, then direction cosines are l = cos α, m = cos β and n = cos γ.

Given that α = 90°, β = 135° and γ = 45°. Thus, the direction cosines of the given line is

l = cos 90°, m = cos 135° and n = cos 45°.

or, l = 0, m = -1/√2, n = 1/√2.

**NCERT Solutions for Class 12 Maths Three Dimensional Geometry ****Exercise**** 11.1 Ques No 2.**

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

**NCERT Solutions:**

If a line makes angles α, β and γ with x, y and z-axes, then direction cosines are l = cos α, m = cos β and n = cos γ.

And, *l*^{2} + *m*^{2} + *n*^{2} = 1 …(1)

Given that α = β = γ. Thus, the direction cosines of the given line is

*l* = cos α, *m* = cos α and *n* = cos α.

or, *l* = *m* = *n*

or, *l*^{2} + *l*^{2} + *l*^{2} = 1

or, 3*l*^{2} = 1

or, *l* = ±1/√3

Thus, *l* = *m* = *n *= ±1/√3.

Therefore, the direction cosines of the given line are (1/√3, 1/√3, 1/√3) or (-1/√3, -1/√3, -1/√3).

**NCERT Solutions for Class 12 Maths Three Dimensional Geometry ****Exercise**** 11.1 Ques No 3.**

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

**NCERT Solutions:**

Let a, b, c be direction ratios of a line, then the direction cosines of the line are

Given that the direction ratios of the line is a = –18, b = 12, c = –4. Then,

Thus, the direction cosines of the line are

**NCERT Solutions for Class 12 Maths Three Dimensional Geometry ****Exercise**** 11.1 Ques No 4.**

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

**NCERT Solutions:**

The direction ratios of line joining A and B are (– 1 – 2), (– 2 – 3), (1 – 4) Or, – 3, – 5, –3.

The direction ratios of line joining B and C are (5 + 1), (8 + 2), (7 – 1) Or, 6, 10, 6

It is clear that direction ratios of AB and BC are proportional.

Hence, AB is parallel to BC.

But point B is common to both AB and BC. Therefore, A, B, C are collinear points.

**NCERT Solutions for Class 12 Maths Three Dimensional Geometry ****Exercise**** 11.1 Ques No 5.**

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

**NCERT Solutions:**