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

**Ex 11.1 Class 12 Maths Question 1.**

If a line makes angles 90°, 135°, 45° with the and z axes respectively, find its direction cosines.**Solution:**

Direction angles are 90°, 135°, 45°

Direction cosines are

**Ex 11.1 Class 12 Maths Question 2.**

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

**Solution:**

Let direction angle be α each

∴ Direction cosines are cos α, cos α, cos α

But l² + m² + n² = 1

∴cos² a + cos² a + cos² a = 1

**Ex 11.1 Class 12 Maths Question 3.**

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

Now given direction ratios of a line are -8,12,-4

∴ a = -18,b = 12,c = -4

Direction cosines are

**Ex 11.1 Class 12 Maths Question 4.**

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

**Solution:**

Let the points be A(2,3,4), B (-1, -2,1), C (5,8,7).

Let direction ratios of AB be

**Ex 11.1 Class 12 Maths Question 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:**

The vertices of triangle ABC are A (3, 5, -4), B (-1,1,2), C (-5, -5, -2)

(i) Direction ratios of AB are (-4,-4,6)

Direction cosines are