# 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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