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, (x1, y1, z1) and (x2, y2, z2), are given by, x2 − x1, y2 − 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)
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
