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

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

Three Dimensional Geometry Exercise 11.1 Solutions

1. If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.

Solution

Let direction cosines of the line be l, m, and n. 
I = cos 90° = 0 
m = cos 135° = -1/√2 
n = cos 45° = 1/√2 
Therefore, the direction cosines of the line are 0, -1/√2  and 1/√2.


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

Solution

Let the direction cosines of the line make an angle a with each of the coordinate axes. 
∴ I = cos α, m = cos α, n = cos α
l2 + m2 + n2 = 1  
⇒ cos2 α + cosα + cos2 α = 1 
⇒ 3 cos2 α = 1
⇒ cos2 α = 1/3
⇒ cos α = ± 1/√3
Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are ± 1/√3, ± 1/√3, and ± 1/√3.


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

Solution

If a line has direction ratios of -18, 12 and -4, then its direction cosines are:

i.e., -18/22 , 12/22, -4/22 
-9/11, 6/11, -2/11 
Thus, the direction cosines are -9/11, 6/11, and -2/11.


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

Solution

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.


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 Δ ABC 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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