Question

Find the vector equation of the line through $A(3,4,-7)$ and $B(6,-1,1)$. Also find the cartesian form.

Answer 1

$\overrightarrow{a}=(3,4,-7)=3\hat{i}+4\hat{j}-7\hat{k}$

$\overrightarrow{b}=(1,-1,6)=\hat{i}-\hat{j}+6\hat{k}$

$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$

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

$\overrightarrow{r}=3\hat{i}+4\hat{j}-7\hat{k}+\lambda (-2\hat{i}-5\hat{j}+13\hat{k})$

$z\hat{i}+y\hat{j}+z\hat{k}=(3-2\lambda )\hat{i}+(4-5\lambda )\hat{j}+(-7+13\lambda )\hat{k}$

$x=3-2\lambda$, $y=4-5\lambda$, $z=-7+13\lambda$

Solve for $\lambda$,

$\frac{x-3}{-2}=\frac{y-4}{-5}=\frac{3+7}{13}$