## NCERT Solutions for Class 12 Maths Exercise 11.1 Chapter 11 Three Dimensional Geometry – FREE PDF Download

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 (Ex 11.1) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.1 Questions with Solutions to help you to revise complete Syllabus and Score More marks.

# NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry (Ex 11.1) Exercise 11.1

**1.If a line makes angles with the and axes respectively, find its direction cosines.**

**Ans.**Here and

Since direction cosines of a line making angles with the and axes respectively are

Therefore, the direction cosines of the required line are:

cos90∘=0;cos90∘=0;cos135∘=−12√;cos135∘=−12;cos45∘=12√cos45∘=12

**2.Find the direction cosines of a line which makes equal angles with the co-ordinate axes.**

**Ans.**Let a line make equal angles with the co-ordinate axes.

Direction cosines of the line are ……….(i)

Putting in eq. (i), direction cosines of the required line making equal angles with the co-ordinate axes are

Direction cosines of a line making equal angles with the co-ordinate axes in the positive i.e., first octant are

**3.If a line has direction ratios , then what are its direction cosines?**

**Ans.**We know that if are direction ratios of a line, then direction cosines of the line are:

……….(i)

Here direction ratios of the line are

Putting the values in eq. (i),

Hence, direction cosines of required line are .

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

**Ans.**The given points are A (2, 3, 4), B and C (5, 8, 7)

Direction ratios of the line joining A and B are

…….(i)

= (say)

Again Direction ratios of the line joining B and C are

= (say)……….(ii)

From eq. (i) and (ii),

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

**5.Find the direction cosines of the sides of the triangle whose vertices are and **

**Ans.**Direction ratios of the line joining A and B are

Direction cosines of line AB are

Now Direction ratios of the line joining B and C are

Direction cosines of line BC are

⇒−416+36+16√,−616+36+16√,−416+36+16√⇒−416+36+16,−616+36+16,−416+36+16

Direction ratios of the line joining C and A are

Direction cosines of line CA are