2017 H2 Maths Paper 1, Question 6

$\mathbf{r} = \mathbf{a} + t\mathbf{b}$ is the vector equation of a straight line: it passes through the point with position vector $\mathbf{a}$, and is parallel to (has direction) $\mathbf{b}$. As the parameter $t$ varies over all real numbers, $\mathbf{r}$ traces out every point on this line.

$\mathbf{r} \cdot \mathbf{n} = d$ is the vector equation of a plane with unit normal vector $\mathbf{n}$: it consists of all points whose position vector $\mathbf{r}$ has the same projection onto $\mathbf{n}$, namely $d$.

Since $\mathbf{n}$ is a unit vector, $d = \mathbf{r} \cdot \mathbf{n}$ is the length of the projection of $\mathbf{r}$ onto $\mathbf{n}$ for any point on the plane, i.e. $d$ is the perpendicular distance from the origin to the plane.

Substituting $\mathbf{r} = \mathbf{a} + t\mathbf{b}$ into $\mathbf{r} \cdot \mathbf{n} = d$:

Since $\mathbf{b} \cdot \mathbf{n} \neq 0$, we may divide by it:

Substituting back into $\mathbf{r} = \mathbf{a} + t\mathbf{b}$:

Since $\mathbf{b} \cdot \mathbf{n} \neq 0$, the line is not parallel to the plane, so it meets the plane at exactly one point.