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CBSE Class 12 Math 2011 Solved Paper

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Question : 21
Total: 29
Find the angle between the following pair of lines:
−x+2
−2
 =
y−1
7
 =
z+3
−3
 and
x+2
−1
 =
2y−8
4
 =
z−5
4
And check whether the lines are parallel or perpendicular.
Solution:  
Let
→
b1
 and
→
b2
 be the vector parallel to the pair to lines,
−x+2
−2
 =
y−1
7
 =
z+3
−3
 and
x+2
−1
 =
2y−8
4
 =
z−5
4
 , respectively.
Now,
−x+2
−2
 =
y−1
7
 =
z+3
−3
 ⇒
x−2
2
 =
y−1
7
 =
z+3
−3
x+2
−1
 =
2y−8
4
 =
z−5
4
⇒
x+2
−1
 =
y−4
2
 =
z−5
4
∴
→
b1
 = 2
^
i
+7
^
j
−3
^
k
 and
→
b2
 = −
^
i
+2
^
j
+4
^
k
|
→
b1
| = √(2)2+(7)2+(−3)2 = √62
|
→
b2
| = √(−1)2+(2)2+(4)2 = √21
→
b1
.
→
b2
 =
(2
^
i
+7
^
j
−3
^
k
).(−
^
i
+2
^
j
+4
^
k
)

= 2 (- 1) + 7 × 2 + (- 3) . 4
= - 2 + 14 - 12
= 0
The angle θ between the given pair of lines is given by the relation,
cos θ = |
→
b1
.
→
b2
|
→
b1
||
→
b2
|
|
⇒ cos θ =
0
√62×√21
 = 0
⇒ θ = cos−1 (0) =
π
2
Thus, the given lines are perpendicular to each other and the angle between them is 90°.
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