Vectors in 3 Dimensions (Cambridge International A-Level Mathematics): Exam Questions

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

$\overset{\rightarrow}{OA} = 3 i - 4 j + 2 k$

$\overset{\rightarrow}{OB} = - 5 i + 2 j - 8 k$

Find the exact length of $A B$.

In the triangle $A B C$, the vectors $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{AC}$ are given by

$\overset{\rightarrow}{AB} = i + 4 j - 2 k$

$\overset{\rightarrow}{AC} = 6 i - 2 j + 8 k$

(i) Find the vector $\overset{\rightarrow}{BC}$.

(ii) Hence, or otherwise, find the length of $B C$.

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

$\overset{\rightarrow}{OP} = 2 i - j - 5 k$

$\overset{\rightarrow}{OQ} = - 6 i - 12 j + 11 k$

(i) Find the vector $\overset{\rightarrow}{PQ}$.

(ii) Find the length of $P Q$.

Find a unit vector parallel to $\overset{\rightarrow}{PQ}$.

With respect to the origin $O$, the point $A$ has position vector $\overset{\rightarrow}{OA} = - 3 i + 4 j + 9 k$.

The vectors $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{CD}$ are given by

$\overset{\rightarrow}{AB} = 4 i - 5 j + 2 k$

$\overset{\rightarrow}{CD} = - 2 i + \frac{5}{2} j - k$

Find the position vector of $B$.

State whether or not the vectors $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{CD}$ are parallel, giving a reason for your answer.

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

$\overset{\rightarrow}{OP} = 4 i - 3 j + 12 k , \overset{\rightarrow}{OQ} = 3 i - 7 j + 9 k , \overset{\rightarrow}{OR} = 7 i - 9 j + 15 k$

Find the lengths $P Q$, $P R$ and $Q R$. Hence determine whether the triangle $P Q R$ is scalene, isosceles or equilateral.

The vectors $p$ and $q$ are defined by

\mathbf{p} = 14 \mathbf{i} + \left(a + b\right) \mathbf{j} + \left(c - b + 1\right) \mathbf{k}

$q = a i + 6 j - 4 k$

Given that $p = 2 q$, find the values of $a$, $b$ and $c$.

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

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

Find $\overset{\rightarrow}{OF}$ and $\overset{\rightarrow}{OG}$ in terms of $a$, $b$ and $c$.

(i) Explain why $\overset{\rightarrow}{GE} = a$.

(ii) Show that $\overset{\rightarrow}{OF} + \overset{\rightarrow}{FE} = \overset{\rightarrow}{OG} + \overset{\rightarrow}{GE}$. Hence state $\overset{\rightarrow}{OE}$ in terms of $a$, $b$ and $c$.

It is now given that $\overset{\rightarrow}{OA} = 5 i$.

Find the exact length of $O E$.

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

\overrightarrow{OA} = \begin{pmatrix} - 2 \\ 5 \\ - 7 \end{pmatrix}

\overrightarrow{OB} = \begin{pmatrix} - 7 \\ k \\ 3 \end{pmatrix}

Given that $A B = 5 \sqrt{14}$, 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} = \begin{pmatrix} 12 \\ 3 \\ - 3 \end{pmatrix} , \quad \overrightarrow{OQ} = \begin{pmatrix} 7 \\ - 8 \\ k \end{pmatrix} , \quad \overrightarrow{OR} = \begin{pmatrix} 3 \\ 3 \\ - 12 \end{pmatrix}

Given that the triangle $P Q R$ is equilateral, find the value of $k$.

The vectors $a$ and $b$ are defined by

$a = - 12 i - 7 j + 15 k$

\mathbf{b} = 4 p \mathbf{i} + \left(p q r + 2 q r - p\right) \mathbf{j} - p q \mathbf{k}

Given that $a = b$, 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 $a$, $b$ and $c$ are $\overset{\rightarrow}{OA}$, $\overset{\rightarrow}{OB}$ and $\overset{\rightarrow}{OC}$ respectively.

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

Let $P$ be a point on $O E$, and let $Q$ be a point on $A G$.

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

$\overset{\rightarrow}{OP} = λ \overset{\rightarrow}{OE}$

$\overset{\rightarrow}{OQ} = a + μ \overset{\rightarrow}{AG}$

where $λ$ and $μ$ are constants with $0 \leq λ \leq 1$ and $0 \leq μ \leq 1$.

By solving the equation $\overset{\rightarrow}{OP} = \overset{\rightarrow}{OQ}$, using your results from parts (a) and (b), show that the diagonals $O E$ and $A G$ 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} = \begin{pmatrix} - 1 \\ 3 \\ 14 \end{pmatrix}

\overrightarrow{OB} = \begin{pmatrix} 2 k \\ 3 k \\ 13 \end{pmatrix}

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

The vectors $a$ and $b$ are defined by

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

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

Given that $a = b$, 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} = \begin{pmatrix} 2 \\ - 3 \\ 1 \end{pmatrix} , \quad \overrightarrow{OQ} = \begin{pmatrix} - 1 \\ - 4 \\ 3 \end{pmatrix} , \quad \overrightarrow{OR} = \begin{pmatrix} k \\ 0 \\ 3 \end{pmatrix}

Given that the triangle $P Q R$ 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 $a$, $b$ and $c$ are $\overset{\rightarrow}{OA}$, $\overset{\rightarrow}{OB}$ and $\overset{\rightarrow}{OC}$ respectively.

Using vector methods, show that the diagonals $C D$ and $B F$ bisect each other.

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

\overrightarrow{OA} = \begin{pmatrix} 5 \\ 0 \\ - 1 \end{pmatrix}

\overrightarrow{OB} = \begin{pmatrix} k \\ - 2 \\ 3 k \end{pmatrix}

Given that $A B = 6 \sqrt{k}$ and that $O B = \sqrt{14}$, find the value of $k$.

The vectors $a$ and $b$ are defined by

$a = p q i - 24 j + 9 r k$

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

Given that $a = 3 b$ 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} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} , \quad \overrightarrow{OB} = \begin{pmatrix} - 8 \\ 10 \\ 1 \end{pmatrix} , \quad \overrightarrow{OC} = \begin{pmatrix} - 3 \\ 6 \\ - 10 \end{pmatrix} , \quad \overrightarrow{OD} = \begin{pmatrix} - 2 k \\ 13 \\ k \end{pmatrix}

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 $O E$ that divides $O E$ in the ratio $a : b$, where $a > b$.

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