Question

Write vector and the cartesian equations of the lines that passes through the origin and $(5,-2,3)$

Answer 1

The line passing through $(0,0,0)$ and $(5,-2,3)$

$\Rightarrow \frac{x-0}{5-0}=\frac{y-0}{-2-0}=\frac{z-0}{3-0}$

$\Rightarrow \frac{x}{5}=\frac{y}{-2}=\frac{z}{3}$ (cartesian form)

Let $\frac{x}{5}=\frac{y}{-2}=\frac{z}{3}=k$

$x=5k,y=-2k,z=3k$

Let $(x,y,z)$ be position vector

$\therefore x\hat{i}+y\hat{j}+z\hat{j}=k(5\hat{i}+(-2)\hat{j}+3\hat{k})$

Vector form : $k(5\hat{i}+(-2)\hat{j}+3\hat{k})$