Question

The distance of the point having position vector $-\hat{i}+2 \hat{j}+6\hat{k}$ from the straight line passing through the point $(2,3, -4)$ and parallel to the vector, $6\hat{i}+3\hat{j}-4\hat{k}$ is

Answer 1

Equation of the line is $\frac{x-2}{6} =\frac{y-3}{3} = \frac{z+4}{-4}$

Let the point M on the line $l$ is $(6 \lambda+2, 3\lambda+3, -4 \lambda -4)$
Direction ratio’s of PM is $(6 \lambda+3, 3\lambda+1, -4 \lambda -10)$
$\Rightarrow (6 \lambda +3) (6) +(3 \lambda+1) (3) + (-4 \lambda -10) (-4)=0$
$\Rightarrow \lambda =-1$
$\Rightarrow M$ is $(-4, 0, 0)$
$\therefore PM = \sqrt{9+4+36}=7$