If the direction ratios of two lines are (1, 2, 3) and (-2, 3, -4), Then the angle between the lines is:

Question

Question

This question was previously asked in

AAI ATC Junior Executive 21 Feb 2023 Shift 1 Official Paper

Answer 1

Given:

The direction ratios of two lines are (1, 2, 3) and (-2, 3, -4)

Concept:

If given the direction ratios of two lines are \(\rm (a_1,b_1,c_1) \ and\ (a_2,b_2,c_2)\) then

\(\rm \cos\theta=|\ ({\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}})|\)

Calculation:

The direction ratios of two lines are (1, 2, 3) and (-2, 3, -4) then

\(\rm a_1=1,b_1=2,c_1=3\ and\ a_2=-2,b_2=3,c_2=-4\)

Let the angle between the two lines be then

\(\rm \cos\theta=|\ ({\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}})|\)

\(\rm \implies \cos\theta=|{\frac{(1)(-2)+(2)(3)+(3)(-4)}{\sqrt{(1)^2+(2)^2+(3)^2}\sqrt{(-2)^2+(3)^2+(-4)^2}}}|\)

\(\rm \implies cos\theta= |{\frac{-8}{\sqrt{14}\sqrt{29}}}|\)

\(\rm \implies cos\theta= |{\frac{-8}{\sqrt{406}}}|\)

\(\rm \implies\theta=cos^{-1} ({\frac{8}{\sqrt{406}}})\)

Hence the option (1) is correct.

Download Solution PDF