Relative to the origin $O$, the distinct points $A$, $B$, $C$ and $D$ have position vector

Question

Relative to the origin $O$, the distinct points $A$, $B$, $C$ and $D$ have position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ respectively. It is given that $(\mathbf{a}-\mathbf{b})\times(\mathbf{a}-\mathbf{c})=\mathbf{0}$. State the relationship between the points $A$, $B$ and $C$, justifying your answer.[2] It is given that $\mathbf{b}$ is a unit vector. Give a geometrical interpretation of $\left|\mathbf{b}.(\mathbf{c}-\mathbf{d})\right|$.[1] It is also given that $\mathbf{a}=\begin{pmatrix}2\ -3\ 4\end{pmatrix}$, $\mathbf{b}=\begin{pmatrix}p\ q\ q\end{pmatrix}$, where $p$ and $q$ are constants, and that $OA$ is perpendicular to $OB$. Find the possible values of $p$ and $q$.[4]

Accepted Answer

(a) $A$, $B$ and $C$ are collinear, because $\overrightarrow{BA}$ and $\overrightarrow{CA}$ are non-zero parallel vectors with the common point $A$ (b) The length of the projection of $CD$ onto a line parallel to $\mathbf{b}$ (c) $p=\dfrac{1}{3},\ q=-\dfrac{2}{3}$ or $p=-\dfrac{1}{3},\ q=\dfrac{2}{3}$

N2026 P1 Q3

N2026 P1 Q3