AHL 3.11—Vector equation of a line in 2d and 3d

The point $\mathbf{a}$ is the fixed point on the line, and $\mathbf{b}$ is the direction vector.

Using $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}$ with $\mathbf{a}=\left(\begin{array}{c}4\\ -2\\ 1\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}3\\ 1\\ -2\end{array}\right)$.

$\mathbf{r}=\left(\begin{array}{c}4\\ -2\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}3\\ 1\\ -2\end{array}\right)$.

This matches option A exactly.

Swaps the roles of $\mathbf{a}$ and $\mathbf{b}$, giving a line through the wrong point with the wrong direction.

Reverses the order of the direction vector's components incorrectly, changing the direction to an unrelated vector.

Uses the point itself as the direction vector, which is a modelling error since direction must be parallel to the line, not the position vector.

Only option A correctly anchors at $C$ and moves along the given direction.

The direction vector is $\overrightarrow{PQ}=\mathbf{q}-\mathbf{p}$.

$5-(-1)=6$.

$-2-3=-5$.

$\overrightarrow{PQ}=\left(\begin{array}{c}6\\ -5\end{array}\right)$, matching option A.

This is $\overrightarrow{QP}$, the reverse direction, from subtracting in the wrong order.

Adds coordinates instead of subtracting, an arithmetic slip unrelated to the direction vector definition.

Correct x-component but sign error on the y-component, likely from mishandling the negative in $-2-3$.

Only option A correctly computes $Q-P$ componentwise.