Relative to the origin $O$, the points $A$ and $B$ have non-zero and non-parallel position vec

Question

Relative to the origin $O$, the points $A$ and $B$ have non-zero and non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The plane $p$ has equation $\mathbf{r} \cdot \mathbf{n} = 0$. Given that $\mathbf{a} \cdot \mathbf{n} = \mathbf{b} \cdot \mathbf{n} \neq 0$, show that $\vec{AB}$ is perpendicular to $\mathbf{n}$. Hence, describe the geometrical relationship between $\vec{AB}$ and the plane $p$.[2] Write down the equation of a line parallel to $\vec{AB}$ that is contained in the plane $p$.[1] The point $F$ is the foot of perpendicular from a point $C$ with position vector $\mathbf{c}$ to the plane $p$. Find the position vector of point $F$, giving your answer in terms of $\mathbf{c}$ and $\mathbf{n}$.[3]

Accepted Answer

(a) $(\mathbf b-\mathbf a)\cdot\mathbf n=0$, so $AB$ is parallel to the plane (shown). (b) $\mathbf r=\lambda(\mathbf b-\mathbf a)$, $\lambda\in\mathbb R$. (c) $\overrightarrow{OF}=\mathbf c-\frac{\mathbf c\cdot\mathbf n}{\mathbf n\cdot\mathbf n}\mathbf n$.

2025 ACJC P1 Q5

2025 ACJC P1 Q5

Solution:

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