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Learn Exercise 11.1 with Free Lessons & Tips

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 l, m, and n.

Therefore, the direction cosines of the line are

Comments

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 α, m = cos α, n = cos α

Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are

Comments

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.

Comments

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, (x1, y1, z1) and (x2, y2, z2), are given by, x2x1, y2y1, and z2z1.

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.

Comments

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

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

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