# Equation of a Line in Vector Form

Here you will learn equation of a line in vector form passing through a fixed point and passing through two points.

Let’s begin –

## Equation of a Line in Vector Form

The vector equation of a straight line passing through a fixed point with position vector $$\vec{a}$$ and parallel to a given vector $$\vec{b}$$ is

$$\vec{r}$$ = $$\vec{a}$$ + $$\lambda \vec{b}$$, where $$\lambda$$ is scalar.

Note : In the above equation $$\vec{r}$$ is the position vector of any point P (x, y, z) on the line. Therefore, $$\vec{r}$$ = $$x\hat{i} + y\hat{j} + z\hat{k}$$.

Example : Find the vector equation of a line which passes through the point with position vector $$2\hat{i} – \hat{j} + 4\hat{k}$$ and is in the direction $$\hat{i} + \hat{j} – 2\hat{k}$$.

Solution : Here $$\vec{a}$$ = $$2\hat{i} – \hat{j} + 4\hat{k}$$ and $$\vec{b}$$ = $$\hat{i} + \hat{j} – 2\hat{k}$$.

So, the vector equation of the required line is

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

or, $$\vec{r}$$ = ($$2\hat{i} – \hat{j} + 4\hat{k}$$) + $$\lambda (\hat{i} + \hat{j} – 2\hat{k})$$, where $$\lambda$$ is a scalar.

#### Equation of Line in Vector Form Passing Through Two Points

The vector equation of line passing through two points with position vectors $$\vec{a}$$ and $$\vec{b}$$ is

$$\vec{r}$$ = $$\lambda$$ $$(\vec{b} – \vec{a})$$, where $$\lambda$$ is a scalar

Example : 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.