1.

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

Answer»

The CORRECT answer is (c) \(θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)

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

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

\(θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)


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