Vectors in 2 Dimensions (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are given by

$\mathbf{a}=\left(\begin{array}{c}3\\ -p\end{array}\right),\mathbf{b}=\left(\begin{array}{c}p\\ 4\end{array}\right),\mathbf{c}=\left(\begin{array}{c}9\\ 3\end{array}\right)$

where $p$ is a constant.

Given that $\mathbf{a}+\mathbf{b}$ is parallel to $\mathbf{c}$, find the value of $p$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are given by

$\mathbf{a}=\left(\begin{array}{c}7\\ 2\end{array}\right),\mathbf{b}=\left(\begin{array}{c}m\\ -3\end{array}\right),\mathbf{c}=\left(\begin{array}{c}5\\ n\end{array}\right)$

Given that $\mathbf{a}+2\mathbf{b}=\mathbf{c}$, find the values of $m$ and $n$.

The vector $\mathbf{d}$ is given by

$\mathbf{d}=\left(\begin{array}{c}-5\\ k\end{array}\right)$

Given that $\left|\mathbf{d}\right|=15$, find the two possible values of $k$, giving your answers as exact values.

The point $A$ lies on the line with equation $y=3x+5$. With respect to the origin $O$, the position vector of $A$ is $\overrightarrow{OA}=2k\mathbf{i}+7k\mathbf{j}$, where $k$ is a constant.

Find the value of $k$, and hence determine the coordinates of $A$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are given by

$\mathbf{a}=\left(\begin{array}{c}-5\\ 17\end{array}\right),\mathbf{b}=\left(\begin{array}{c}k\\ 5\end{array}\right),\mathbf{c}=\left(\begin{array}{c}9\\ -29\end{array}\right)$

where $k$ is a constant.

Given that $\mathbf{a}-\mathbf{b}$ is parallel to $\mathbf{b}+\mathbf{c}$, find the value of $k$.

The vector $\overrightarrow{AB}$ is given by $\overrightarrow{AB}=11\mathbf{i}-2\mathbf{j}$.

Find

(i) the magnitude of $\overrightarrow{AB}$, giving your answer as an exact value,

(ii) the angle between $\overrightarrow{AB}$ and the positive $x$-axis, giving your answer in degrees correct to two decimal places.

Find a unit vector in the direction of $\overrightarrow{AB}$.

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

$\overrightarrow{OA}=-4\mathbf{i}-7\mathbf{j},\overrightarrow{OB}=3\mathbf{j},\overrightarrow{OC}=6\mathbf{i}+18\mathbf{j}$

Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

Show that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel, and state what this tells you about the points $A$, $B$ and $C$.

In the triangle $ABC$, $\overrightarrow{AB}=\mathbf{a}$ and $\overrightarrow{AC}=\mathbf{b}$. The point $P$ divides $BC$ in the ratio $3:2$.

(i) Write down $\overrightarrow{BC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

(ii) Find $\overrightarrow{BP}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Given that $\mathbf{a}=7\mathbf{i}+8\mathbf{j}$ and $\mathbf{b}=12\mathbf{i}+3\mathbf{j}$, find $\overrightarrow{BP}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are given by

$\mathbf{a}=\left(\begin{array}{c}-1\\ n\end{array}\right),\mathbf{b}=\left(\begin{array}{c}5\\ -4\end{array}\right),\mathbf{c}=\left(\begin{array}{c}m\\ 6\end{array}\right)$

where $m$ and $n$ are constants.

Given that the resultant of $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ is the zero vector, find the values of $m$ and $n$.

The vector $\mathbf{d}$ is given by

$\mathbf{d}=\left(\begin{array}{c}-3k\\ k\end{array}\right)$

Given that $\left|\mathbf{d}\right|=2\sqrt{15}$, find the two possible values of $k$, giving your answers as exact values.

In the triangle $ABC$, $\overrightarrow{AB}=5\mathbf{i}+8\mathbf{j}$ and $\overrightarrow{BC}=\mathbf{i}-5\mathbf{j}$.

Explain why $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\mathbf{0}$.

Find $\overrightarrow{CA}$ and calculate its magnitude.

The point $A$ lies on the curve with equation $y=x^2-2$. With respect to the origin $O$, the position vector of $A$ is $\overrightarrow{OA}=3k\mathbf{i}-17k\mathbf{j}$, where $k$ is a positive constant.

Find the value of $k$, and hence determine the coordinates of $A$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are given by

$\mathbf{a}=\left(\begin{array}{c}3\\ 5\end{array}\right),\mathbf{b}=\left(\begin{array}{c}-3k\\ k\end{array}\right),\mathbf{c}=\left(\begin{array}{c}0\\ -4\end{array}\right)$

where $k$ is a constant.

Given that $\mathbf{a}-\mathbf{b}$ is parallel to $\mathbf{a}+\mathbf{c}$, find the value of $k$.

The vector $\overrightarrow{AB}$ has a magnitude of $6\sqrt{3}$ and makes an angle of 150° with the positive $x$-axis.

Find $\overrightarrow{AB}$ in the form $x\mathbf{i}+y\mathbf{j}$, where both $x$ and $y$ are given as exact values.

Find a unit vector in the direction of $\overrightarrow{AB}$.

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

$\overrightarrow{OA}=-9\mathbf{i}+4\mathbf{j},\overrightarrow{OB}=-6\mathbf{i},\overrightarrow{OC}=3\mathbf{i}-12\mathbf{j}$

Use a vector method to show that the points $A$, $B$ and $C$ lie on the same straight line.

$A$, $B$ and $C$ are the three vertices of a triangle. $\overrightarrow{AC}=5\mathbf{i}-2\mathbf{j}$ and $\overrightarrow{BC}=-3\mathbf{i}+k\mathbf{j}$, where $k$ is a constant.

Find $\overrightarrow{AB}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $k$.

Given that $\left|\overrightarrow{AB}\right|=\sqrt{89}$, find the two possible values of $k$.

The point $A$ lies on the circle with equation $(x-11{)}^2+(y-7{)}^2=34$. With respect to the origin $O$, the position vector of $A$ is $\overrightarrow{OA}=3k\mathbf{i}+5k\mathbf{j}$, where $k$ is a constant.

Find the value of $k$, and hence determine the coordinates of $A$.

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

$\overrightarrow{OA}=-6\mathbf{i}-2\mathbf{j},\overrightarrow{OB}=\mathbf{i}+m\mathbf{j},\overrightarrow{OC}=3\mathbf{i}-8\mathbf{j}$

where $m$ is a constant.

Given that $A$, $B$ and $C$ lie on the same straight line, use a vector method to find the value of $m$.

The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are given by

$\mathbf{a}=\left(\begin{array}{c}8\\ m\end{array}\right),\mathbf{b}=\left(\begin{array}{c}n\\ -2\end{array}\right),\mathbf{c}=\left(\begin{array}{c}m\\ n\end{array}\right)$

where $m$ and $n$ are constants.

Given that $\mathbf{a}+\mathbf{b}=\mathbf{c}-2\mathbf{b}$, find the values of $m$ and $n$.

The vector $\mathbf{d}$ is given by

$\mathbf{d}=\left(\begin{array}{c}2k+1\\ 2k-1\end{array}\right)$

Given that $\left|\mathbf{d}\right|=3k\sqrt{2}$, find the value of $k$, justifying your answer. Give your answer as an exact value.

The vector $\overrightarrow{AB}$ has a magnitude of $2\sqrt{6}$ and makes an angle of 165° with the positive $y$-axis, measuring anticlockwise from the positive $y$-axis.

Find $\overrightarrow{AB}$ in the form $a\mathbf{i}+b\mathbf{j}$, where both $a$ and $b$ are given as exact values.

Find a unit vector in the direction of $\overrightarrow{AB}$.

In the triangle $ABC$, $D$ is the midpoint of $AB$ and $E$ is the midpoint of $AC$.

$BE$ and $CD$ intersect at the point $F$.

Given that $\overrightarrow{AB}=2\mathbf{a}$ and $\overrightarrow{AC}=2\mathbf{b}$, write the vectors $\overrightarrow{BC}$, $\overrightarrow{BE}$ and $\overrightarrow{CD}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

By setting up and solving suitable vector equations, prove that each of $BE$ and $CD$ divides the other in the ratio $1:2$.