How Do You Write the Vector Equation of a Line \(r = a + \lambda b\)?

Write the vector equation of a line by combining the position vector of any point on the line with a scalar multiple of a non-zero direction vector:

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

Here, $\lambda$ varies over all real numbers, generating every point on the line.

The mechanism is translation and scaling. The vector $\lambda\mathbf{b}$ moves from the fixed point represented by $\mathbf{a}$. Positive and negative values of $\lambda$ move in opposite directions, while $\lambda=0$ gives the known point itself. Because $\mathbf{b}\neq\mathbf{0}$, changing the parameter traces the complete line rather than a single point.

Writing the equation from two points

Suppose the line passes through $P(2,-1,3)$ and $Q(5,3,1)$.

First, choose one point to provide the position vector:

$$\mathbf{a}=\begin{pmatrix}2\\-1\\3\end{pmatrix}.$$

Next, calculate a direction vector by subtracting the position vectors:

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

Therefore,

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

To verify the result, substitute $\lambda=0$ to recover $P$, then solve for the parameter that gives $Q$. Here, $\lambda=1$ produces $Q$, since adding to gives . This check also catches reversed subtraction: using is valid, but the corresponding parameter for becomes .

An equivalent equation may use $Q$ as the position vector or any non-zero scalar multiple of $\overrightarrow{PQ}$ as the direction vector. These describe the same geometric line.

A common misconception is to use the second point’s position vector directly as $\mathbf{b}$. The direction must instead be found by subtraction, such as $\overrightarrow{PQ}=\mathbf{q}-\mathbf{p}$.

IB exam technique

This is IB Mathematics AA HL, AHL 3.14, and the analytic method is suitable for Paper 1 as well as Papers 2 and 3. For a command such as find, show the subtraction used to obtain the direction vector before stating the final equation; a correct unsupported equation may not earn all method marks.