Question

If the equations of two lines L1 and L2 are \(\vec{r}=\vec{a_1}+λ\vec{b_1}\) and \(\vec{r}=\vec{a_2}+μ\vec{b_2}\), then which of the following is the correct formula for the angle between the two lines?

(a) cos⁡θ=\(\left |\frac{\vec{a_1}.\vec{a_2}}{|\vec{b_1}||\vec{a_2}|}\right |\)

(b) cos⁡θ=\(\left |\frac{\vec{a_1}.\vec{a_2}}{|\vec{a_1}||\vec{a_2}|}\right |\)

(c) cos⁡θ=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\)

(d) cos⁡θ=\(\left |\frac{\vec{a_1}.\vec{b_2}}{|\vec{a_1}||\vec{b_2}|}\right |\)

I got this question at a job interview.

Query is from Three Dimensional Geometry topic in section Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct option is (c) cos⁡θ=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\)

The best explanation: Given that the equations of the lines are

\(\vec{r}=\vec{a_1}+λ\vec{b_1} \,and \,\vec{r}=\vec{a_2}+μ\vec{b_2}\)

∴ the angle between the two lines is given by

cos⁡θ=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\).