Dividing vectors into ratios | Maths | Quizalize & Oak National Academy

Starter quiz

Work out the resultant vector for ${10 \choose -2} + {4 \choose 9}$ .

${2 \choose 7}$

${14 \choose 7}$ ✓

${5\choose 7}$

${9\choose 7}$
Match each notation to its correct meaning.

$\overrightarrow{OX}$

⇔

A vector starting from $O$ and ending at $X$ ✓

$OX$

⇔

A line segment with points $O$ and $X$ at each end ✓

$\overrightarrow{XO}$

⇔

A vector starting from $X$ and ending at $O$ ✓

$\overrightarrow{XY}$

⇔

A vector starting from $X$ and ending at $Y$ ✓

$XY$

⇔

A line segment with points $X$ and $Y$ at each end ✓
Match each vector to an equivalent vector statement.

${12 \choose -9}$

⇔

$3\times {4 \choose -3}$ ✓

${16 \choose -12}$

⇔

$4\times {4 \choose -3}$ ✓

${-18 \choose -6}$

⇔

$2\times {-9 \choose -3}$ ✓

${4\choose -20}$

⇔

$-2\times {-2 \choose 10}$ ✓

${-27\choose -27}$

⇔

$-3\times {9 \choose 9}$ ✓
Match each vector statement to an equivalent vector statement.

$5\mathbf{a}+2\mathbf{b}$

⇔

$\frac{1}{2}(10\mathbf{a}+4\mathbf{b})$ ✓

$4\mathbf{a}+2\mathbf{b}$

⇔

$\frac{1}{3}(12\mathbf{a}+6\mathbf{b})$ ✓

$2\mathbf{a}+3\mathbf{b}$

⇔

$\frac{1}{5}(10\mathbf{a}+15\mathbf{b})$ ✓

$5\mathbf{a}-2\mathbf{b}$

⇔

$\frac{1}{2}(10\mathbf{a}-4\mathbf{b})$ ✓

$15\mathbf{a}-12\mathbf{b}$

⇔

$\frac{3}{4}(20\mathbf{a}-16\mathbf{b})$ ✓
$Select the vector path for <Math>\overrightarrow{AD}</Math>.$

Select the vector path for $\overrightarrow{AD}$ .

$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}$

$-\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}$ ✓

$\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{CD}$
Select the statements which are true.

$\frac{1}{2}\overrightarrow{IH}=\overrightarrow{CD}$ ✓

$2\overrightarrow{CD}=\overrightarrow{IH}$ ✓

$-\overrightarrow{IH}=\overrightarrow{HG}$

$-\overrightarrow{AB}=\overrightarrow{FE}$

$\frac{1}{2}\overrightarrow{FE}=\overrightarrow{BC}$ ✓

Exit quiz

X is a point on a line AB. Match each vector statement to the correct ratio.

$AX:XB \; \text{is} \; 1:4$

⇔

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

$AX:XB \; \text{is} \; 1:3$

⇔

$\frac{1}{4}\overrightarrow{AB}=\overrightarrow{AX}$ ✓

$AX:XB \; \text{is} \; 1:1$

⇔

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

$AX:XB \; \text{is} \; 4 : 1$

⇔

$\frac{4}{5}\overrightarrow{AB}=\overrightarrow{AX}$ ✓

$AX:XB \; \text{is} \; 2 : 3$

⇔

$\frac{2}{5}\overrightarrow{AB}=\overrightarrow{AX}$ ✓
B is a point on the line OL. Match each ratio to the correct vector statement.

$OB : BL \; \text{is} \; 1 : 5$

⇔

$\frac{1}{6}\overrightarrow{OL}=\overrightarrow{OB}$ ✓

$OB : BL \; \text{is} \; 1 : 1$

⇔

$\frac{1}{2}\overrightarrow{OL}=\overrightarrow{OB}$ ✓

$OB : BL \; \text{is} \; 11 : 1$

⇔

$\frac{11}{12}\overrightarrow{OL}=\overrightarrow{OB}$ ✓

$OB : BL \; \text{is} \; 1 : 11$

⇔

$\frac{1}{12}\overrightarrow{OL}=\overrightarrow{OB}$ ✓
O, A and B are in a straight line. You are given that $\overrightarrow{OA}=12\mathbf{a}+20\mathbf{b}$ and OA : AB is in the ratio 2 : 3. Work out vector $\overrightarrow{AB}$ .

$6\mathbf{a}+10\mathbf{b}$

$24\mathbf{a}+40\mathbf{b}$

$18\mathbf{a}+30\mathbf{b}$ ✓

$7.2\mathbf{a}+12\mathbf{b}$

$8\mathbf{a}+13\mathbf{b}$
You are given that O, A and B are in a straight line, $\overrightarrow{OA}=12\mathbf{a}+20\mathbf{b}$ and OB : BA is in the ratio 2 : 3. Work out vector $\overrightarrow{BA}$ .

$6\mathbf{a}+10\mathbf{b}$

$24\mathbf{a}+40\mathbf{b}$

$18\mathbf{a}+30\mathbf{b}$

$7.2\mathbf{a}+12\mathbf{b}$ ✓

$8\mathbf{a}+13\mathbf{b}$
XYZ is a triangle. $\overrightarrow{XY}= 5\mathbf{a}-7.5\mathbf{b}$ and N lies on $\overrightarrow{XY}$ such that $\overrightarrow{XN}=3\mathbf{a}-4.5\mathbf{b}$ . Write the ratio of XN : NY.

1 : 2

2 : 3

3 : 2 ✓

1 : 3
In triangle OAB, $\overrightarrow{OA}=18\mathbf{a}$ , $\overrightarrow{OB}=12\mathbf{b}$ . P is on AB. AP:PB = 1:2. $\overrightarrow{OP}=k(6\mathbf{a}+2\mathbf{b})$ . The value of $k$ is ______.

'2' ✓

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Lesson Details

Key learning points

A vector can be divided into parts using a ratio.
By converting the ratio to a fraction, you can express each part in terms of the original vector.
This allows you to perform further manipulations.

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

Vector - A vector can be used to describe a translation.
Displacement - Displacement is the distance from the starting point when measured in a straight line.
Resultant vector - A resultant vector is the single vector that produces the same effect as a combination of other vectors.