Geometric proofs with vectors | Maths | Quizalize & Oak National Academy

Starter quiz

Work out the magnitude of the vector ${4 \choose -3}$ .

'5' ✓
Which of the following vectors are parallel to $2\mathbf{a}-3\mathbf{b}$ ?

$4\mathbf{a}-1\mathbf{b}$

$6\mathbf{a}-4\mathbf{b}$

$4\mathbf{a}-6\mathbf{b}$ ✓

$-10\mathbf{a}+15\mathbf{b}$ ✓
Which of the following vectors are parallel to ${8\choose -12}$ ?

${4\choose -6}$ ✓

${-4\choose 6}$ ✓

${-3\choose2}$

${24\choose 36}$

${0.8\choose -1.2}$ ✓
X is a point on the line AB. Match each vector statement to the correct ratio.

AX : XB is 1 : 4

⇔

$\frac{1}{5}\overrightarrow{AB}=\overrightarrow{AX}$ ✓

AX : XB is 3 : 8

⇔

$\frac{3}{11}\overrightarrow{AB}=\overrightarrow{AX}$ ✓

AX : XB is 1 : 1

⇔

$\frac{1}{2}\overrightarrow{AB}=\overrightarrow{AX}$ ✓

AX : XB is 3 : 1

⇔

$\frac{3}{4}\overrightarrow{AB}=\overrightarrow{AX}$ ✓
Match each value of $x$ to the vector equation that it satisfies.

$x = 13$

⇔

${x \choose -2} + {12 \choose 7} = {25 \choose 5}$ ✓

$x=12$

⇔

${x\choose -7} + {-7 \choose -7} = {5 \choose -14}$ ✓

$x=4$

⇔

$x{2 \choose 8} = {8 \choose 32}$ ✓

$x=8$

⇔

$3{3\choose 4} + {3 \choose -4} = {12 \choose x}$ ✓
OACB is a parallelogram. M is the midpoint of OA and N is the midpoint of OB. $\overrightarrow{OM}=2\mathbf{a}$ and $\overrightarrow{ON}=2\mathbf{b}$ .

$\frac{1}{2}\overrightarrow{AB}=\overrightarrow{MN}$ ✓

$\overrightarrow{MN}=2\overrightarrow{AB}$

$\frac{1}{3}\overrightarrow{AB}=\overrightarrow{MN}$

$2\overrightarrow{MN}=\overrightarrow{AB}$ ✓

$\overrightarrow{AB}=\frac{1}{2}\overrightarrow{MN}$

Exit quiz

Three points that lie along the same straight line are said to be ______.

'collinear' ✓
Select the correct statements to show that $ODEF$ is a trapezium.

$\overrightarrow{OD}=2\overrightarrow{EF}$

$\overrightarrow{OD}=2\overrightarrow{FE}$ ✓

$\overrightarrow{OD}=\frac{1}{2}\overrightarrow{FE}$

$\overrightarrow{OF}=k\overrightarrow{ED}$

$\overrightarrow{OF}\neq k\overrightarrow{ED}$ ✓
Select the correct statements to show that $OAEF$ is not a parallelogram.

$\overrightarrow{OF}=\overrightarrow{AE}$

$\overrightarrow{OF}=2\overrightarrow{AE}$

$2\overrightarrow{OF}=\overrightarrow{AE}$ ✓

$\overrightarrow{OA}=\overrightarrow{FE}$

$\overrightarrow{OA}\neq k\overrightarrow{FE}$ ✓
$<Math>ABCD</Math> is a kite. <Math>W,X,Y,Z</Math> are midpoints of the lines they lie on. <Math>\overrightarrow{XW} = k(\mathbf b - \mathbf a)</Math> where <Math>k= </Math> <span class="blank">______</span>$

$ABCD$ is a kite. $W,X,Y,Z$ are midpoints of the lines they lie on. $\overrightarrow{XW} = k(\mathbf b - \mathbf a)$ where $k=$ ______

'2' ✓
$<Math>ABCD</Math> is a kite. <Math>W,X,Y,Z</Math> are midpoints of the lines they lie on. Find <Math>\overrightarrow{ZW}</Math>.$

$ABCD$ is a kite. $W,X,Y,Z$ are midpoints of the lines they lie on. Find $\overrightarrow{ZW}$ .

$- \mathbf c - \mathbf b$

$-2 \mathbf c -2 \mathbf b$ ✓

$2\mathbf c + 2\mathbf b$

$\mathbf c + \mathbf b$
$<Math>ABCD</Math> is a kite. <Math>W,X,Y,Z</Math> are midpoints of the lines they lie on. Find <Math>\overrightarrow{XZ}</Math>.$

$ABCD$ is a kite. $W,X,Y,Z$ are midpoints of the lines they lie on. Find $\overrightarrow{XZ}$ .

$- \mathbf c - 2\mathbf b + \mathbf a$

$\mathbf c + 2\mathbf b - \mathbf a$

$-2 \mathbf c -4 \mathbf b + 2\mathbf a$

$2\mathbf c + 4\mathbf b - 2\mathbf a$ ✓

Worksheet

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Lesson Details

Key learning points

Your knowledge of the properties of polygons can be applied to vector problems.
A regular polygon has a relationship between sides that can be represented with vector notation.
Your knowledge can be used to identify vectors and multiples of known vectors.
Through manipulation, you can prove certain properties.

Common misconception

When calculating the resultant vector, pupils can incorrectly sum vectors due to opposite directions or proportions of vectors.

Encourage pupils to write a clear vector pathway, sometimes using highlighters can help visualise this pathway.

Keywords

Collinear - When three or more points lie on a single straight line, these points are said to be collinear.