Question

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector $2\hat{i}-\hat{j}+4\hat{k}$ and is in the direction $\hat{i}+2\hat{j}-\hat{k}$.

Answer 1

It is given that the line passes through the point with position vector $a=2\hat{i}-\hat{j}+4\hat{k}andb=\hat{i}+2\hat{j}-\hat{k}$
It is known that a line through a point with position vector a and parallel to b i.e., in direction of b is given by the equation, r = a + $\lambda$b.
$\therefore r=2\hat{i}-\hat{j}+4\hat{k}+\lambda (\hat{i}+2\hat{j}-\hat{k})$
This is the required equation of the line in vector form.
Let $r=x\hat{i}+y\hat{j}+z\hat{k}\therefore x\hat{i}+y\hat{j}+z\hat{k}=(\lambda +2)\hat{i}+(2\lambda -1)\hat{j}+(-\lambda +4)\hat{k}$
Eliminating $\lambda$, we obtain the Cartesian form equation as
$\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-4}{-1}$
This is the required equation of the given line in Cartesian form.