If L1 and L2 have the direction ratios a_1,b_1,c_1 \,and \,a_2,b_2,c_2 respectively then what is the angle between the lines?

Category: QuestionsIf L1 and L2 have the direction ratios a_1,b_1,c_1 \,and \,a_2,b_2,c_2 respectively then what is the angle between the lines?
Editor">Editor Staff asked 11 months ago

If L1 and L2 have the direction ratios a_1,b_1,c_1 \,and \,a_2,b_2,c_2
respectively then what is the angle between the lines?
 
(a) θ=tan^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |
 
 
(b) θ=2tan^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |
 
 
(c) θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |
 
 
(d) θ=2 \,cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |
 
 
The question was asked in an interview for internship.
 
Query is from Three Dimensional Geometry in chapter Three Dimensional Geometry of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct answer is (c)
θ=co
s
−1




a
1
a
2
+
b
1
b
2
+
c
1
c
2
a
2
1
+
b
2
1
+
c
2
1










a
2
2
+
b
2
2
+
c
2
2














 
 
For explanation I would say: If L1 and L2 have the direction ratios
a
1
,
b
1
,
c
1
and
a
2
,
b
2
,
c
2
respectively then the angle between the lines is given by
 
cosθ=




a
1
a
2
+
b
1
b
2
+
c
1
c
2
a
2
1
+
b
2
1
+
c
2
1










a
2
2
+
b
2
2
+
c
2
2














 
 
θ=co
s
−1




a
1
a
2
+
b
1
b
2
+
c
1
c
2
a
2
1
+
b
2
1
+
c
2
1










a
2
2
+
b
2
2
+
c
2
2