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Exam-Style Questions on Vectors

Problems on Vectors adapted from questions set in previous exams.

1.

IGCSE Extended

OABC is a parallelogram with O as origin. The position vector of A is \(a\) and the position vector of C is \(c\).

F is the mid-point of AB and the point E divides the line OC such that OE:EC = 2:1.

The point E also divides the line AD such that AE:ED = 3:2.

Find the following in terms of \(a\) and \(c\).

(a) \(\overrightarrow{OB}\)

(b) \(\overrightarrow{AC}\)

(c) \(\overrightarrow{AE}\)

(d) the position vector of F.

(e) \(\overrightarrow{AD}\)

(f) \(\overrightarrow{BD}\)


2.

GCSE Higher

(a) Shape \(A\) is translated to shape \(B\) using the vector \( \begin{pmatrix}m\\n\\ \end{pmatrix}\). What are the values of \(m\) and \(n\)?

Vectors

(b) Vectors \(a, b, c, d\) and \(e\) are drawn on an isometric grid. Write each of the vectors \(c, d\) and \(e\) in terms of \(a\) and/or \(b\).

Vectors

3.

GCSE Higher

ABCD is a quadrilateral. The points E, F, G and H are the midpoints of the sides of this quadrilateral.

Right-angled triangle $$\vec{CH} = a, \vec{DG} = b \text{ and } \vec{DF} = c$$

Show that HG is parallel to EF.


4.

IGCSE Extended

ABCDOE is a regular hexagon with O as origin. The position vector of A is \(a\) and the position vector of B is \(b\).

Find the following in terms of \(a\) and \(b\).

(a) \(\overrightarrow{BA}\)

(b) \(\overrightarrow{OE}\)

(c) the position vector of C.

If the sides of the hexagon are each of length 10cm calculate:

(d) the size of angle \(BCD\).

(e) the area of triangle \(BCD\).

(f) the length of the line from B to D.

(g) the area of the hexagon.


5.

GCSE Higher

Consider a triangle ABC where M is the midpoint of AB and F is the point on BC where BF:FC = 3:4.

Triangle

(a) If \(\overrightarrow{AB}=b\) and \(\overrightarrow{AC}=c\) work out \(\overrightarrow{MF}\) in terms of \(b\) and \(c\) giving your answer in its simplest form.

(b) Use your answer to part (a) to explain whether MF is parallel to AC or not.


6.

IB Standard

The line \(L_1\) passes through the points A(3, 5, 1) and B(3, 6, 0).

(a) Show that \(AB=\begin{pmatrix} 0 \\ 1 \\ -1 \\ \end{pmatrix}\)

(b) Find a direction vector for \(L_1\)

(c) a vector equation for \(L_1\)

Another line \(L_2\) has equation \(\begin{pmatrix} 8 \\ 3 \\ -2 \\ \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 0 \\ \end{pmatrix}\). The lines \(L_1\) and \(L_2\) intersect at the point P.

(d) Find the coordinates of P.

(e) Find a direction vector for \(L_2\).

(f) Hence, find the angle between \(L_1\) and \(L_2\).


7.

IB Standard

Consider two perpendicular vectors \(p\) and \(q\).

(a) Let \(r=p-q\). Draw a diagram to show what this relationship might look like.

(b) If \(p=\begin{pmatrix} 4 \\ 1 \\ -3 \\ \end{pmatrix}\) and \(q=\begin{pmatrix} 3 \\ n \\ -5 \\ \end{pmatrix}\), where \(n\in \mathbb Z\), find \(n\).


8.

IB Standard

The line L is parallel to the vector \(\begin{pmatrix} 2 \\ 5 \\ \end{pmatrix} \),

(a) Find the gradient of the line L .

The line L passes through the point (11, 3).

(b) Write down the equation of the line L in the form \(y=ax+b\)

(c) Find a vector equation for the line L.


9.

IGCSE Extended

(a) If A is the point (3,5) write down the position vector of A.

(b) If B is the point (6,9) find \(\mid\overrightarrow{AB} \mid\) the magnitude of \( \overrightarrow{AB}\).

The following diagram is not to scale.

Vectors

\(O\) is the origin, \(\overrightarrow{OP}=p\) and = \(\overrightarrow{OQ}=q\).

\(OP\) is extended to \(R\) so that \(OP=PR\).

\(OQ\) is extended to \(S\) so that \(OQ=QS\).

(c) Write down \(\overrightarrow{RQ}\) in terms of \(p\) and \(q\).

(d) \(PS\) and \(RQ\) intersect at \(M\) and \(RM=2 MQ\).

Use vectors to find the ratio \(PM:PS\), showing all your working.


10.

GCSE Higher
Sine and Cosine Rules

In the diagram above (not drawn to scale) \(X\) is the point on \(AB\) such that \(AX:XB = 9:4\).

The position vector of \(A\) is \(3a\) and the position vector of \(B\) is \(3b\).

Find the value of \(k\) if \(\vec{OX} = k(4a + 9b)\) where \(k\) is a scalar quantity.


11.

IB Standard

George and Hugo like to fly model airplanes. On one day George's plane takes off from level ground and shortly after that Hugo's plane takes off.

The position of George’s plane \(s\) seconds after it takes off is given by \(\begin{pmatrix} 1 \\ 2 \\ 0 \\ \end{pmatrix} + s\begin{pmatrix} 5 \\ -2 \\ 6 \\ \end{pmatrix} \) where the distances are in metres.

(a) Find the speed of George’s plane to the nearest integer.

(b) Find the height of George’s plane after four seconds.

The position of Hugo’s airplane \(t\) seconds after it takes off is given by \(\begin{pmatrix} 4 \\ -4 \\ 0 \\ \end{pmatrix}+t\begin{pmatrix} 7 \\ -2 \\ 9 \\ \end{pmatrix} \) where the distances are in metres.

(c) Show that the paths of the planes are not perpendicular.

The two airplanes collide at the point \((46, -16, 54)\).

(d) How long after George’s plane takes off does Hugo’s plane take off ?


The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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