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

#### Question 1:

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

Let direction cosines of the line be lm, and n.
Therefore, the direction cosines of the line are

#### Question 2:

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

Let the direction cosines of the line make an angle α with each of the coordinate axes.
∴ l = cos αm = cos αn = cos α
Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are

#### Question 3:

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

If a line has direction ratios of −18, 12, and −4, then its direction cosines are
Thus, the direction cosines are.

#### Question 4:

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

The given points are A (2, 3, 4), B (− 1, − 2, 1), and C (5, 8, 7).
It is known that the direction ratios of line joining the points, (x1y1z1) and (x2y2z2), are given by, x2 − x1y2 − y1, and z2 − z1.
The direction ratios of AB are (−1 − 2), (−2 − 3), and (1 − 4) i.e., −3, −5, and −3.
The direction ratios of BC are (5 − (− 1)), (8 − (− 2)), and (7 − 1) i.e., 6, 10, and 6.
It can be seen that the direction ratios of BC are −2 times that of AB i.e., they are proportional.
Therefore, AB is parallel to BC. Since point B is common to both AB and BC, points A, B, and C are collinear.

#### 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)