Parallel vectors in algebraic vector notation | Maths

Starter quiz

Identify which of these statements are true.

Parallel vectors have the same gradient. ✓

Parallel vectors must be in the same direction.

Parallel vectors can have different magnitudes. ✓

Parallel vectors may have opposite directions. ✓

All parallel vectors have the same length.
Which of the following vectors are parallel to ${3 \choose 5}$ ?

${1.5 \choose 4}$

${6 \choose 10}$ ✓

${12 \choose 20}$ ✓

${5 \choose7}$

${1.5\choose 2.5}$ ✓
$Which of the following vectors is parallel to <Math>\overrightarrow{IA}</Math>?$

Which of the following vectors is parallel to $\overrightarrow{IA}$ ?

$\overrightarrow{ED}$

$\overrightarrow{BC}$

$\overrightarrow{AB}$

$\overrightarrow{HG}$ ✓
$Here is a regular hexagon with two labelled vectors. The centre of the hexagon is O. Select the vectors which are parallel to <Math>\overrightarrow{OX}</Math>.$

Here is a regular hexagon with two labelled vectors. The centre of the hexagon is O. Select the vectors which are parallel to $\overrightarrow{OX}$ .

$\overrightarrow{QW}$

$\overrightarrow{QR}$ ✓

$\overrightarrow{RZ}$

$\overrightarrow{WX}$ ✓

$\overrightarrow{XW}$ ✓
Which of the following vectors are parallel to ${2 \choose -3}$ ?

${-3 \choose 2}$

${-8 \choose 12}$ ✓

${1 \choose -1.5}$ ✓

${6 \choose -18}$

${-6 \choose 9}$ ✓
Match each column vector to its gradient.

${8\choose 4}$

⇔

Gradient = $\frac{1}{2}$ ✓

${4 \choose 8}$

⇔

Gradient = $2$ ✓

${-4 \choose 8}$

⇔

Gradient = $-2$ ✓

${8 \choose -4}$

⇔

Gradient = $-\frac{1}{2}$ ✓

Exit quiz

Select the correct statements for parallel vectors. Parallel vectors __________.

are scalar multiples of one another ✓

are not collinear

can be in opposite directions ✓

can only be positive

have the same gradient ✓
Which of these vectors are parallel to ${4 \choose -3}$ ?

${-8 \choose 6}$ ✓

${5 \choose -2}$

${40 \choose -30}$ ✓

${2\choose -1.5}$ ✓

${6 \choose -7}$
Which of these vectors are parallel to $3\mathbf{a}+5\mathbf{b}$ ?

$6\mathbf{a}+10\mathbf{b}$ ✓

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

$12\mathbf{a}+20\mathbf{b}$ ✓

$6\mathbf{a}+11\mathbf{b}$

$30\mathbf{a}+50\mathbf{b}$ ✓
Match each vector to a vector that is parallel to it.

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

⇔

$-2\mathbf{a}-3\mathbf{b}$ ✓

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

⇔

$6\mathbf{a}+4\mathbf{b}$ ✓

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

⇔

$-8\mathbf{a}+12\mathbf{b}$ ✓

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

⇔

$-4\mathbf{a}+6\mathbf{b}$ ✓
OACB is a parallelogram. M is the midpoint of OA and N is the midpoint of OB. $\overrightarrow{OM}=10\mathbf{a}$ and $\overrightarrow{ON}=10\mathbf{b}$ . Select the correct scalar multipliers.

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

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

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

$3\overrightarrow{MN}=\overrightarrow{AB}$
ABCD is a kite. AX:XB = 1:1, BY:YC = 1:1, DZ:ZC = 1:1 and AW:WD=1:1, $\overrightarrow{AW}= 3\mathbf{a}$ , $\overrightarrow{AX}=3\mathbf{b}$ and $\overrightarrow{DZ}=3\mathbf{c}$ .

$\overrightarrow{XY} = \overrightarrow{WZ}$ ✓

$\overrightarrow{YC} = \overrightarrow{ZD}$

$\overrightarrow{AX} = \overrightarrow{AW}$

$\overrightarrow{XW} = \overrightarrow{YZ}$ ✓

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Lesson Details

Key learning points

Parallel vectors have the same gradient.
In algebraic form, this may be seen when one vector is a multiple of another.
Vectors with opposite signs are parallel but act in opposite directions.

Common misconception

Parallel vectors can only be in the same direction.

Two vectors are parallel if one can be written as a scalar multiple of the other. This scalar multiple can be negative therefore the direction can be opposite, but the gradients remain equivalent.

Keywords

Parallel - Two lines are parallel if they are straight lines that are always the same (non-zero) distance apart.