Vectors in 2D (Edexcel International A Level (IAL) Maths: Pure 4): Exam Questions

In triangle ABC,  $\overrightarrow{AB}=7\mathbf{i}+\mathbf{j}$ and $\overrightarrow{AC}=4\mathbf{i}-3\mathbf{j}$

(i) Write down $\overrightarrow{CA}$in terms of i and j.     

(ii) Find $\overrightarrow{BC}$.

Calculate $\left|\overrightarrow{BC}\right|$.

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

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

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

Given that $\left|\mathbf{d}\right|=15$, find two possible values for k.  Give your answer as an exact value.

The point A lies on the line with equation$y=3x+5$.  The position vector of A is$\overrightarrow{OA}=2k\mathbf{i}+7k\mathbf{j}$.  Find the value of k, and hence determine the coordinates of A.

The vectors a, b and c are given as

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

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

$\overrightarrow{AB}=11\mathbf{i}-2\mathbf{j}$

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 $\overrightarrow{AB}$.

A ship leaves its starting position O in port and travels 300 km on a bearing of 120°.  It then travels 500 km due south before dropping anchor at point A.  Given that the position vector of A relative to O is $\left(x\mathbf{i}+y\mathbf{j}\right)$km, find the exact values of x and y.

Two forces F₁ and F₂ act on a particle, where  ${\mathbf{F}}_1=7\mathbf{i}-2\mathbf{j}$newtons and ${\mathbf{F}}_2=-12\mathbf{i}-10\mathbf{j}$ newtons.

The resultant force R acting on the particle is given by $\mathbf{R}={\mathbf{F}}_1+{\mathbf{F}}_2$.

Calculate the magnitude of R in newtons.

A third force ${\mathbf{F}}_3=k\mathbf{j}$ newtons is to be applied to the particle.  The constant k is to be selected so that the line of action of the new resultant force ${\mathbf{R}}_{\mathrm{new}}={\mathbf{F}}_1+{\mathbf{F}}_2+{\mathbf{F}}_3$is at an angle of 45 degrees to the vector j, measured anticlockwise.

Find the value of k.

Points A, B and C have position vectors $\overrightarrow{OA}=-4\mathbf{i}-7\mathbf{j}$,  $\overrightarrow{OB}=3\mathbf{j}$  and  $\overrightarrow{\mathrm{OC}}=6\mathbf{i}+18\mathbf{j}$, respectively.

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

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

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

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

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

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

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

Given that the resultant of a, b and c is the zero vector, find the values of m and n. 

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

Given that $\left|\mathbf{d}\right|=2\sqrt{15},$ find two possible values for k.  Give your answer as an exact value.

The point A lies on the curve with equation $y=x^2-2$.  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 a, b and c are given as

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

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

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 $\overrightarrow{AB}$.

In the enchanted kingdom of Vectoria, a magical flying unicorn takes off from the wizard’s palace at the point known as O and travels 30 km on a bearing of 300°.  Chased by an evil dragon, it then travels an unknown distance of  k km due north before reaching the enchanted grove at point P.  Given that the position vector of P relative to O is $\left(x\mathbf{i}+y\mathbf{j}\right)\mathrm{km}$, and that the straight-line distance between the grove and the palace is known to be $30\sqrt{3}\mathrm{km}$, find the exact values of x and y.

Two forces F₁ and F₂ act on a particle, where  ${\mathbf{F}}_1=5\mathbf{i}-3\mathbf{j}$newtons and ${\mathbf{F}}_2=x\mathbf{i}+y\mathbf{j}$ newtons.

The resultant force R acting on the particle is given by $\mathbf{R}={\mathbf{F}}_1+{\mathbf{F}}_2$, and acts in a direction parallel to the vector $\left(-\mathbf{i}-3\mathbf{j}\right)$.

Find the angle between R and the vector j, giving your answer in degrees correct to 2 decimal places.

Show that $3x-y=-18$.

Given that$y=-3$, find the magnitude of R.

Points A, B and C have position vectors $\overrightarrow{OA}=-9\mathbf{i}+4\mathbf{j}$,  $\overrightarrow{OB}=-6\mathbf{i}$ and  $\overrightarrow{OC}=3\mathbf{i}-12\mathbf{j},$ respectively.

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

In triangle ABC, point F lies on AB and point G lies on BC.  

F divides AB in the ratio m:n.  

The line segment FG is parallel to AC.

Explain why $\overrightarrow{BG}=\lambda \overrightarrow{BC}$ for some constant $\lambda$, where $0\le \lambda \le 1$.

Given that $\overrightarrow{AB}=\mathbf{a}$  and $\overrightarrow{AC}=\mathbf{b}$,   show that

$\overrightarrow{FG}=\left(\frac{n}{m+n}-\lambda \right)\mathbf{a}+\lambda \mathbf{b}$

Using your result from (b), prove that G divides BC in the ratio n:m.

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 i, j and k.

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

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

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

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

Given that $\left|\mathbf{d}\right|=3k\sqrt{2}$, find two possible values for k.  Give your answer as an exact value.

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

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

Explain why a line passing through O and A must be a tangent to the circle.

Points A, B and C have position vectors $\overrightarrow{OA}=-6\mathbf{i}-2\mathbf{j}$,  $\overrightarrow{OB}=\mathbf{i}+m\mathbf{j}$and  $\overrightarrow{OC}=3\mathbf{i}-8\mathbf{j}$, respectively.

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

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 $\overrightarrow{AB}$.

A ship is searching for a radio buoy whose transmitter has ceased functioning.  The ship sets out from point O and heads in the approximate direction of the buoy, travelling at a constant speed of 40 km/h in a direction parallel to the vector $\mathbf{i}+3\mathbf{j}$.  After travelling for ninety minutes the ship has reached point P.  At that time, the ship receives a brief transmission from the buoy indicating that the buoy is at a bearing of  210° from the ship.  The ship heads on that bearing at the same constant speed, and reaches the buoy at point Q in another 45 minutes.  Given that vector $\overrightarrow{OQ}=x\mathbf{i}+y\mathbf{j}\mathrm{km}$, find the exact values of x and y.

Given that vector $\overrightarrow{OQ}=x\mathbf{i}+y\mathbf{j}\mathrm{km}$, find the exact values of x and y.

How far was the buoy from the ship, and at what bearing, at the time the ship initially left point O?  Give the distance in kilometers, and give your answers correct to 1 decimal place.

In an experiment, three forces are acting on a particle.  ${\mathbf{F}}_1=7\mathbf{i}-\mathbf{j}$newtons and ${\mathbf{\text{F}}}_2=x\mathbf{i}+y\mathbf{j}$ newtons are both constant forces, although the values of x and y are initially unknown.  The third force is${\mathbf{F}}_3=k\mathbf{i}+k\sqrt{3}\mathbf{j}$newtons, where $k\ge 0$ is a parameter that can be varied by the experimenters.  The resultant force R acting on the particle is given by $\mathbf{R}={\mathbf{F}}_1+{\mathbf{F}}_2+{\mathbf{F}}_3$.

Given that $\mathbf{R}=\mathbf{0}$ when the magnitude of ${\mathbf{F}}_3$ is 10 newtons, find the exact values of x and y.

Find the magnitude of ${\mathbf{F}}_2$ and the angle it makes with the vector i.  Give your answers correct to 1 decimal place.

In triangle ABC, D is the midpoint of AB and E is the midpoint of AC.  BE and CD intersect at 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 a and b.

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