Find the vector equation of a line which passes through the point A (3, 4, -7) and B (1, -1, 6)

Solution :

We know that the vector equation of line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is,

$\vec{r}$ = $\lambda$ $(\vec{b} – \vec{a})$

Here $\vec{a}$ = $3\hat{i} + 4\hat{j} – 7\hat{k}$ and $\vec{b}$ = $\hat{i} – \hat{j} + 6\hat{k}$.

So, the vector equation of the required line is

$\vec{r}$ = ($3\hat{i} + 4\hat{j} – 7\hat{k}$) + $\lambda$  ($\hat{i} – \hat{j} + 6\hat{k}$ – $3\hat{i} + 4\hat{j} – 7\hat{k}$)

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

where $\lambda$ is a scalar.

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