How Do You Write the Vector Equation of a Line?

To write the vector equation of a line, use a known point on the line and a non-zero vector parallel to it:

$$\mathbf{r}=\mathbf{a}+\lambda\mathbf{b},\qquad \lambda\in\mathbb{R}.$$

Here, $\mathbf{a}$ is the position vector of a point on the line and $\mathbf{b}$ is its direction vector.

The Mechanism

The position vector $\mathbf{a}$ fixes one point on the line. The direction vector $\mathbf{b}$ controls the line's direction, while the scalar parameter $\lambda$ generates every point on the line. The same structure applies in both two and three dimensions.

For example, suppose the line passes through $A(1,2,-1)$ and $B(4,-2,5)$. Its direction vector is

$$\overrightarrow{AB}=\begin{pmatrix}4\\-2\\5\end{pmatrix}-\begin{pmatrix}1\\2\\-1\end{pmatrix}=\begin{pmatrix}3\\-4\\6\end{pmatrix}.$$

Therefore,

$$\mathbf{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}3\\-4\\6\end{pmatrix},\qquad \lambda\in\mathbb{R}.$$

This may also be written in parametric form:

$$x=1+3\lambda,\qquad y=2-4\lambda,\qquad z=-1+6\lambda.$$

You can verify the equation using the original points. When $\lambda=0$, the equation gives $A(1,2,-1)$. When $\lambda=1$, it gives $B(4,-2,5)$. This confirms that both points lie on the line and that the direction vector moves from to .

The equation is not unique. Any point on the line can be used as the position vector, and any non-zero scalar multiple of the direction vector represents the same direction. For example, $\begin{pmatrix}6\\-8\\12\end{pmatrix}$ is valid, although the zero vector is never a valid direction vector.

A common misconception is to use the coordinates of the second point as the direction vector. The direction must be found by subtracting the two position vectors in a consistent order. Reversing the subtraction is acceptable because it only reverses the direction.

Exam Technique

This is HL-only content from AHL 3.11, applicable across GDC-active HL papers. For the command term find, show the vector subtraction before stating the equation, include $\lambda\in\mathbb{R}$, and check that suitable parameter values produce the given points. Do not reject an equivalent equation merely because it uses a different point or a scalar multiple of the direction vector.