Vectors in 3 Dimensions (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

With respect to the origin $O$, the points $A$ and $B$ have position vectors given by

$\overrightarrow{OA}=\left(\begin{array}{c}-2\\ 5\\ -7\end{array}\right)$

$\overrightarrow{OB}=\left(\begin{array}{c}-7\\ k\\ 3\end{array}\right)$

Given that $AB=5\sqrt{14}$, find the possible values of $k$.

With respect to the origin $O$, the points $P$, $Q$ and $R$ have position vectors given by

$\overrightarrow{OP}=\left(\begin{array}{c}12\\ 3\\ -3\end{array}\right),\overrightarrow{OQ}=\left(\begin{array}{c}7\\ -8\\ k\end{array}\right),\overrightarrow{OR}=\left(\begin{array}{c}3\\ 3\\ -12\end{array}\right)$

Given that the triangle $PQR$ is equilateral, find the value of $k$.

The vectors $\mathbf{a}$ and $\mathbf{b}$ are defined by

$\mathbf{a}=-12\mathbf{i}-7\mathbf{j}+15\mathbf{k}$

$\mathbf{b}=4p\mathbf{i}+\left(pqr+2qr-p\right)\mathbf{j}-pq\mathbf{k}$

Given that $\mathbf{a}=\mathbf{b}$, find the values of $p$, $q$ and $r$.

The diagram below shows a cube whose vertices are $O$, $A$, $B$, $C$, $D$, $E$, $F$ and $G$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OC}$ respectively.

Find $\overrightarrow{OE}$ and $\overrightarrow{AG}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.

Let $P$ be a point on $OE$, and let $Q$ be a point on $AG$.

Explain why the vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ can be expressed in the forms

$\overrightarrow{OP}=\lambda \overrightarrow{OE}$

$\overrightarrow{OQ}=\mathbf{a}+\mu \overrightarrow{AG}$

where $\lambda$ and $\mu$ are constants with $0\le \lambda \le 1$ and $0\le \mu \le 1$.

By solving the equation $\overrightarrow{OP}=\overrightarrow{OQ}$, using your results from parts (a) and (b), show that the diagonals $OE$ and $AG$ intersect each other, and determine the ratios into which they are cut by the point of intersection.

With respect to the origin $O$, the points $A$ and $B$ have position vectors given by

$\overrightarrow{OA}=\left(\begin{array}{c}-1\\ 3\\ 14\end{array}\right)$

$\overrightarrow{OB}=\left(\begin{array}{c}2k\\ 3k\\ 13\end{array}\right)$

Given that $AB=\sqrt{163}$ and that $k$ is an integer, find the value of $k$.

The vectors $\mathbf{a}$ and $\mathbf{b}$ are defined by

$\mathbf{a}=\left(p+1\right)\mathbf{i}-7\mathbf{j}+\left(q-3p\right)\mathbf{k}$

$\mathbf{b}=5\mathbf{i}+\left(14q+r\right)\mathbf{j}+\left(1-2r\right)\mathbf{k}$

Given that $\mathbf{a}=\mathbf{b}$, find the values of $p$, $q$ and $r$.

With respect to the origin $O$, the points $P$, $Q$ and $R$ have position vectors given by

$\overrightarrow{OP}=\left(\begin{array}{c}2\\ -3\\ 1\end{array}\right),\overrightarrow{OQ}=\left(\begin{array}{c}-1\\ -4\\ 3\end{array}\right),\overrightarrow{OR}=\left(\begin{array}{c}k\\ 0\\ 3\end{array}\right)$

Given that the triangle $PQR$ is isosceles and that $k>1$, find the value of $k$.

The diagram below shows a cube whose vertices are $O$, $A$, $B$, $C$, $D$, $E$, $F$ and $G$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OC}$ respectively.

Using vector methods, show that the diagonals $CD$ and $BF$ bisect each other.

With respect to the origin $O$, the points $A$ and $B$ have position vectors given by

$\overrightarrow{OA}=\left(\begin{array}{c}5\\ 0\\ -1\end{array}\right)$

$\overrightarrow{OB}=\left(\begin{array}{c}k\\ -2\\ 3k\end{array}\right)$

Given that $AB=6\sqrt{k}$ and that $OB=\sqrt{14}$, find the value of $k$.

The vectors $\mathbf{a}$ and $\mathbf{b}$ are defined by

$\mathbf{a}=pq\mathbf{i}-24\mathbf{j}+9r\mathbf{k}$

$\mathbf{b}=6\mathbf{i}+\left(p-r\right)\mathbf{j}+\left(p-3q\right)\mathbf{k}$

Given that $\mathbf{a}=3\mathbf{b}$ and that $r>0$, find the values of $p$, $q$ and $r$.

$A$, $B$, $C$ and $D$ are the vertices of a regular tetrahedron.

With respect to the origin $O$, the points $A$, $B$, $C$ and $D$ have position vectors given by

$\overrightarrow{OA}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right),\overrightarrow{OB}=\left(\begin{array}{c}-8\\ 10\\ 1\end{array}\right),\overrightarrow{OC}=\left(\begin{array}{c}-3\\ 6\\ -10\end{array}\right),\overrightarrow{OD}=\left(\begin{array}{c}-2k\\ 13\\ k\end{array}\right)$

Find the position vector of $D$.

The diagram below shows a cuboid whose vertices are $O$, $A$, $B$, $C$, $D$, $E$, $F$ and $G$.

$P$ is a point on the diagonal $OE$ that divides $OE$ in the ratio $a:b$, where $a>b$.

Show that if line segment $BP$ is extended it will intersect $FE$, and show that $FE$ is divided in the ratio $\left(a-b\right):b$ by the point of intersection.