Advanced problem solving with vectors

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Lesson details

Learning outcome

I can use my knowledge of vectors to solve problems.

Key learning points

Drawing a diagram can make things clearer.
Adding to an existing diagram can make things clearer.
It can be helpful to start with what you know or can deduce.

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.

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.

Teacher tip

Give pupils the opportunity to calculate a resultant column vector using scalars and summing other vectors. They replace two known values with x and y, thus creating a problem solving question which they could give to a peer to solve. For a greater challenge, replace a third value with z.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

(Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
Three points that lie along the ___________ are said to be collinear.

same curve

same pair of parallel lines

Correct answer: same straight line

Q2.
Select the correct statements to show that $$AEDB$$ is a trapezium.

Correct answer: $$\overrightarrow{AE}=2\overrightarrow{BD}$$

$$\overrightarrow{EA}=2\overrightarrow{BD}$$

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

Correct answer: $$\overrightarrow{AB}\neq k\overrightarrow{ED}$$

$$\overrightarrow{AB}=\overrightarrow{ED}$$

Q3.
Select the correct statements to show that $$ABEF$$ is a parallelogram.

$$\overrightarrow{AF}=\overrightarrow{EB}$$

Correct answer: $$\overrightarrow{AF}=\overrightarrow{BE}$$

$$\overrightarrow{FA}=\overrightarrow{BE}$$

Correct answer: $$\overrightarrow{AB}=\overrightarrow{FE}$$

$$\overrightarrow{AB}=\overrightarrow{EF}$$

Q4.
$$ABCD$$ is a kite. The midpoints of each side are shown. $$\overrightarrow{WX} = k(\mathbf a - \mathbf b)$$ where $$k=$$ .

Correct Answer: 2

Q5.
$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{WZ}$$.

$$- \mathbf c - \mathbf b$$

$$-2 \mathbf c -2 \mathbf b$$

Correct answer: $$2\mathbf b + 2\mathbf c$$

$$\mathbf c + \mathbf b$$

Q6.
$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{ZX}$$.

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

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

Correct answer: $$-2 \mathbf c -4 \mathbf b + 2\mathbf a$$

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

Assessment exit quiz

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6 Questions

Q1.
A vector is the single vector that produces the same effect as a combination of other vectors.

Correct Answer: resultant

Q2.
Given $$\mathbf a = {2 \choose 3}$$ and $$\mathbf b = {-1 \choose 2}$$, find $$4\mathbf a - 2 \mathbf b$$

$$8 \choose8$$

$$8 \choose4$$

$$6 \choose4$$

Correct answer: $$10 \choose8$$

$$10 \choose6$$

Q3.
Given that $$ x\choose 8$$ is parallel to $$ 15 \choose 12$$, then $$x =$$

Correct Answer: 10

Q4.
$$ a {3\choose -2} + 4{ 3 \choose b} = {3\choose 14}$$. The value of $$b$$ is

Correct Answer: 2

Q5.
Given that $$a {2\choose 1} + b{ 4 \choose4}= {16 \choose 14}$$, find the value of $$a + b$$. $$a+b=$$ .

Correct Answer: 5

Q6.
$$\mathbf a = {2\choose -3}$$ and $$\mathbf b = {-2 \choose y}$$. Given that $$3 \mathbf a - 2 \mathbf b$$ is parallel to $$30 \choose -51$$, the value of $$y$$ is .

Correct Answer: 4

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Advanced problem solving with vectors

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Three points that lie along the ___________ are said to be collinear.

Q2.Select the correct statements to show that $$AEDB$$ is a trapezium.

Q3.Select the correct statements to show that $$ABEF$$ is a parallelogram.

Q4.$$ABCD$$ is a kite. The midpoints of each side are shown. $$\overrightarrow{WX} = k(\mathbf a - \mathbf b)$$ where $$k=$$ .

Q5.$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{WZ}$$.

Q6.$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{ZX}$$.

Assessment exit quiz

6 Questions

Q1.A vector is the single vector that produces the same effect as a combination of other vectors.

Q2.Given $$\mathbf a = {2 \choose 3}$$ and $$\mathbf b = {-1 \choose 2}$$, find $$4\mathbf a - 2 \mathbf b$$

Q3.Given that $$ x\choose 8$$ is parallel to $$ 15 \choose 12$$, then $$x =$$

Q4.$$ a {3\choose -2} + 4{ 3 \choose b} = {3\choose 14}$$. The value of $$b$$ is

Q5.Given that $$a {2\choose 1} + b{ 4 \choose4}= {16 \choose 14}$$, find the value of $$a + b$$. $$a+b=$$ .

Q6.$$\mathbf a = {2\choose -3}$$ and $$\mathbf b = {-2 \choose y}$$. Given that $$3 \mathbf a - 2 \mathbf b$$ is parallel to $$30 \choose -51$$, the value of $$y$$ is .

How to plan a lesson using our resources

Q1.
Three points that lie along the ___________ are said to be collinear.

Q2.
Select the correct statements to show that $$AEDB$$ is a trapezium.

Q3.
Select the correct statements to show that $$ABEF$$ is a parallelogram.

Q4.
$$ABCD$$ is a kite. The midpoints of each side are shown. $$\overrightarrow{WX} = k(\mathbf a - \mathbf b)$$ where $$k=$$ .

Q5.
$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{WZ}$$.

Q6.
$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{ZX}$$.

Q1.
A vector is the single vector that produces the same effect as a combination of other vectors.

Q2.
Given $$\mathbf a = {2 \choose 3}$$ and $$\mathbf b = {-1 \choose 2}$$, find $$4\mathbf a - 2 \mathbf b$$

Q3.
Given that $$ x\choose 8$$ is parallel to $$ 15 \choose 12$$, then $$x =$$

Q4.
$$ a {3\choose -2} + 4{ 3 \choose b} = {3\choose 14}$$. The value of $$b$$ is

Q5.
Given that $$a {2\choose 1} + b{ 4 \choose4}= {16 \choose 14}$$, find the value of $$a + b$$. $$a+b=$$ .

Q6.
$$\mathbf a = {2\choose -3}$$ and $$\mathbf b = {-2 \choose y}$$. Given that $$3 \mathbf a - 2 \mathbf b$$ is parallel to $$30 \choose -51$$, the value of $$y$$ is .