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

#### Page No 467:

#### Question 1:

If a line makes angles 90°, 135°, 45° with

*x*,*y*and*z*-axes respectively, find its direction cosines.#### Answer:

Let direction cosines of the line be

*l*,*m*, 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.

#### Answer:

Let the direction cosines of the line make an angle

*α*with each of the coordinate axes.
∴

*l*= cos*α*,*m*= cos*α*,*n*= 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?

#### Answer:

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.

#### Answer:

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, (

*x*_{1},*y*_{1},*z*_{1}) and (*x*_{2},*y*_{2},*z*_{2}), are given by,*x*_{2}−*x*_{1},*y*_{2}−*y*_{1}, and*z*_{2}−*z*_{1}.
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)

#### Answer:

The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2).

The direction ratios of side AB are (−1 − 3), (1 − 5), and (2 − (−4)) i.e., −4, −4, and 6.

Therefore, the direction cosines of AB are

The direction ratios of BC are (−5 − (−1)), (−5 − 1), and (−2 − 2) i.e., −4, −6, and −4.

Therefore, the direction cosines of BC are

-217, -317, -217The direction ratios of CA are 3−(−5), 5−(−5) and −4−(−2) i.e. 8, 10 and -2.

Therefore the direction cosines of CA are

882 + 102 + -22, 1082 + 102 + -22, -282 + 102 + -228242, 10242, -2242442, 542, -142

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