Question

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

Answer 1

The vecctor passes through origin and $(5,-2,3)$

so, the required vector $=(5\hat{i}-2\hat{j}+3\hat{k})-\left(0\right)$

$=5\hat{i}-2\hat{j}+3\hat{k}$

The cartesian equation of the line is given by

$\overrightarrow{r}=\overrightarrow{a}+\lambda (\overrightarrow{b}-\overrightarrow{a})$

$=\lambda (5\hat{i}-2\hat{j}+3\hat{k})$

$\lambda$ is real number.

which is $\frac{x}{5}=\frac{y}{-2}=\frac{z}{3}=\lambda$