Question

Find the angle between the pair of lines \(\frac{x-3}{5}=\frac{y+7}{3}=\frac{z-2}{2} \,and \,\frac{x+1}{3}=\frac{y-5}{4}=\frac{z+2}{8}\).

(a) \(cos^{-1}⁡\frac{43}{\sqrt{3482}}\)

(b) \(cos^{-1}⁡⁡\frac{43}{\sqrt{3382}}\)

(c) \(cos^{-1}⁡⁡\frac{85}{\sqrt{3382}}\)

(d) \(cos^{-1}⁡⁡\frac{34}{\sqrt{3382}}\)

This question was addressed to me in an online interview.

The origin of the question is Three Dimensional Geometry topic in portion Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct answer is (b) \(cos^{-1}⁡⁡\frac{43}{\sqrt{3382}}\)

Easy explanation: The direction ratios are 5, 3, 2 for L1 and 3, 4, 8 for L2

∴ the angle between the two lines is given by

cos⁡θ=\(\frac{(a_1 a_2+b_1 b_2+c_1 c_2)}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\)

=\(\frac{15+12+16}{\sqrt{5^2+3^2+2^2}.\sqrt{3^2+4^2+8^2}}\)

=\(\frac{43}{\sqrt{38}.\sqrt{89}}=\frac{43}{\sqrt{3382}}\)

θ=\(cos^{-1}⁡\frac{43}{\sqrt{3382}}\).