OA = a and OB = b. Work out AB in terms of a and b.

Write down the two vectors the question gives you: OA = a, OB = b Substitute the position vectors: AB = (b) - (a) State the answer: AB = -a + b Answer: AB = -a + b

OA = a and OB = b. Work out BA in terms of a and b.

Write down the two vectors the question gives you: OA = a, OB = b Substitute the position vectors: BA = (a) - (b) State the answer: BA = a - b Answer: BA = a - b

OA = a and OB = b. M is the midpoint of AB. Work out OM in terms of a and b.

Write down the two vectors the question gives you: OA = a, OB = b Write down the position vector of M, the midpoint of AB: OM = (1)/(2)(OA + OB) = (1)/(2)a + (1)/(2)b State the answer: OM = (1)/(2)a + (1)/(2)b Answer: OM = (1)/(2)a + (1)/(2)b

OA = a and OB = b. M is the midpoint of AB. Work out AM in terms of a and b.

Write down the two vectors the question gives you: OA = a, OB = b Write down the position vector of M, the midpoint of AB: OM = (1)/(2)(OA + OB) = (1)/(2)a + (1)/(2)b State the answer: AM = ((1)/(2)a + (1)/(2)b) - (a) = -(1)/(2)a + (1)/(2)b Answer: AM = -(1)/(2)a + (1)/(2)b

Vector geometry and proof Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Vector geometry and proof questions. See exactly how to solve problems on position vectors, triangle law, vectors in terms of a and b, midpoints.

position vectorstriangle lawvectors in terms of a and bmidpointsexact halvesparallelogram

GCSE Higher70 questionsStep-by-step solutions

Question 1

1 markeasy

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

. Work out

\overrightarrow{AB}

in terms of

\mathbf{a}

and

\mathbf{b}

Worked solution

Write down the two vectors the question gives you
$\overrightarrow{OA} = \mathbf{a}, \quad \overrightarrow{OB} = \mathbf{b}$
Every vector in the figure has to be written in terms of $\mathbf{a}$ and $\mathbf{b}$ , so start from the two the question hands you: $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .
Substitute the position vectors
$\overrightarrow{AB} = \left(\mathbf{b}\right) - \left(\mathbf{a}\right)$
$\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OA} = \mathbf{a}$ . The brackets matter: the whole of the second vector is subtracted.
State the answer
$\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}$
So $\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}$ , written in terms of $\mathbf{a}$ and $\mathbf{b}$ as the question asked.

Answer

\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}

Question 2

1 markeasy

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

. Work out

\overrightarrow{BA}

in terms of

\mathbf{a}

and

\mathbf{b}

Worked solution

Write down the two vectors the question gives you
$\overrightarrow{OA} = \mathbf{a}, \quad \overrightarrow{OB} = \mathbf{b}$
Every vector in the figure has to be written in terms of $\mathbf{a}$ and $\mathbf{b}$ , so start from the two the question hands you: $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .
Substitute the position vectors
$\overrightarrow{BA} = \left(\mathbf{a}\right) - \left(\mathbf{b}\right)$
$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ . The brackets matter: the whole of the second vector is subtracted.
State the answer
$\overrightarrow{BA} = \mathbf{a} - \mathbf{b}$
So $\overrightarrow{BA} = \mathbf{a} - \mathbf{b}$ , written in terms of $\mathbf{a}$ and $\mathbf{b}$ as the question asked.

Answer

\overrightarrow{BA} = \mathbf{a} - \mathbf{b}

Question 3

2 markseasy

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

M

is the midpoint of

AB

. Work out

\overrightarrow{OM}

in terms of

\mathbf{a}

and

\mathbf{b}

Worked solution

Write down the two vectors the question gives you
$\overrightarrow{OA} = \mathbf{a}, \quad \overrightarrow{OB} = \mathbf{b}$
Every vector in the figure has to be written in terms of $\mathbf{a}$ and $\mathbf{b}$ , so start from the two the question hands you: $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .
Write down the position vector of M, the midpoint of AB
$\overrightarrow{OM} = \frac{1}{2}\left(\overrightarrow{OA} + \overrightarrow{OB}\right) = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$
The midpoint of $AB$ has position vector the average of the two ends, so $\overrightarrow{OM} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ . Halving is exact — keep it as a fraction.
State the answer
$\overrightarrow{OM} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$
So $\overrightarrow{OM} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ , written in terms of $\mathbf{a}$ and $\mathbf{b}$ as the question asked.

Answer

\overrightarrow{OM} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}

Question 4

2 markseasy

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

M

is the midpoint of

AB

. Work out

\overrightarrow{AM}

in terms of

\mathbf{a}

and

\mathbf{b}

Worked solution

Write down the two vectors the question gives you
$\overrightarrow{OA} = \mathbf{a}, \quad \overrightarrow{OB} = \mathbf{b}$
Every vector in the figure has to be written in terms of $\mathbf{a}$ and $\mathbf{b}$ , so start from the two the question hands you: $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .
Write down the position vector of M, the midpoint of AB
$\overrightarrow{OM} = \frac{1}{2}\left(\overrightarrow{OA} + \overrightarrow{OB}\right) = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$
The midpoint of $AB$ has position vector the average of the two ends, so $\overrightarrow{OM} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ . Halving is exact — keep it as a fraction.
State the answer
$\overrightarrow{AM} = \left(\frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}\right) - \left(\mathbf{a}\right) = -\frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$
So $\overrightarrow{AM} = -\frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ , written in terms of $\mathbf{a}$ and $\mathbf{b}$ as the question asked.

Answer

\overrightarrow{AM} = -\frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}

Question 5

1 markeasy

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

C

is the point such that

OACB

is a parallelogram. Work out

\overrightarrow{OC}

in terms of

\mathbf{a}

and

\mathbf{b}

Worked solution

Write down the two vectors the question gives you
$\overrightarrow{OA} = \mathbf{a}, \quad \overrightarrow{OB} = \mathbf{b}$
Every vector in the figure has to be written in terms of $\mathbf{a}$ and $\mathbf{b}$ , so start from the two the question hands you: $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .
Write down the position vector of C, the fourth vertex of the parallelogram OACB
$\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{OB} \Rightarrow \overrightarrow{OC} = \mathbf{a} + \mathbf{b}$
In a parallelogram $OACB$ the diagonals $OC$ and $AB$ bisect each other, so the two diagonals have the same midpoint and $\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{OB}$ . That gives $\overrightarrow{OC} = \mathbf{a} + \mathbf{b}$ .
State the answer
$\overrightarrow{OC} = \mathbf{a} + \mathbf{b}$
So $\overrightarrow{OC} = \mathbf{a} + \mathbf{b}$ , written in terms of $\mathbf{a}$ and $\mathbf{b}$ as the question asked.

Answer

\overrightarrow{OC} = \mathbf{a} + \mathbf{b}

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