Given a=2i+j+3k and b=i-2j+4k, find a+b.

Write down the given vectors: a=2i+j+3k, b=i-2j+4k Combine the corresponding components: (2+1)i+(1-2)j+(3+4)k=3i-j+7k State the resultant vector: a+b=3i-j+7k Answer: 3i-j+7k

Given a=5i-2k and b=i+3j+k, find a-b.

Write down the given vectors: a=5i-2k, b=i+3j+k Combine the corresponding components: (5-1)i+(0-3)j+(-2-1)k=4i-3j-3k State the resultant vector: a-b=4i-3j-3k Answer: 4i-3j-3k

Given a=i+2j+2k and b=3i-j, find 2a+b.

Write down the given vectors: a=i+2j+2k, b=3i-j Combine the corresponding components: (2+3)i+(4-1)j+(4+0)k=5i+3j+4k State the resultant vector: 2a+b=5i+3j+4k Answer: 5i+3j+4k

Given a=4i+j-k and b=2i+2j+3k, find a-b.

Write down the given vectors: a=4i+j-k, b=2i+2j+3k Combine the corresponding components: (4-2)i+(1-2)j+(-1-3)k=2i-j-4k State the resultant vector: a-b=2i-j-4k Answer: 2i-j-4k

Find the magnitude of the vector a=2i+3j+6k.

Write the vector: a=2i+3j+6k Apply the magnitude formula: |a|=(2)^2+(3)^2+(6)^2 Simplify to state the magnitude: |a|=√(49)=7 Answer: 7

Vectors in three dimensions Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Vectors in three dimensions questions. See exactly how to solve problems on vectors-3d, resultant, magnitude, dot-product.

vectors-3dresultantmagnitudedot-productdistancemidpoint

A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Given

\mathbf{a}=2\mathbf{i}+\mathbf{j}+3\mathbf{k}

and

\mathbf{b}=\mathbf{i}-2\mathbf{j}+4\mathbf{k}

, find

\mathbf{a}+\mathbf{b}

Worked solution

Write down the given vectors
$\mathbf{a}=2\mathbf{i}+\mathbf{j}+3\mathbf{k},\quad \mathbf{b}=\mathbf{i}-2\mathbf{j}+4\mathbf{k}$
List the two vectors in component form.
Combine the corresponding components
$\left(2+1\right)\mathbf{i}+\left(1-2\right)\mathbf{j}+\left(3+4\right)\mathbf{k}=3\mathbf{i}-\mathbf{j}+7\mathbf{k}$
Group the i, j and k components separately.
State the resultant vector
$\mathbf{a}+\mathbf{b}=3\mathbf{i}-\mathbf{j}+7\mathbf{k}$
This is the required resultant vector.

Answer

3\mathbf{i}-\mathbf{j}+7\mathbf{k}

Question 2

2 markseasy

Given

\mathbf{a}=5\mathbf{i}-2\mathbf{k}

and

\mathbf{b}=\mathbf{i}+3\mathbf{j}+\mathbf{k}

, find

\mathbf{a}-\mathbf{b}

Worked solution

Write down the given vectors
$\mathbf{a}=5\mathbf{i}-2\mathbf{k},\quad \mathbf{b}=\mathbf{i}+3\mathbf{j}+\mathbf{k}$
List the two vectors in component form.
Combine the corresponding components
$\left(5-1\right)\mathbf{i}+\left(0-3\right)\mathbf{j}+\left(-2-1\right)\mathbf{k}=4\mathbf{i}-3\mathbf{j}-3\mathbf{k}$
Group the i, j and k components separately.
State the resultant vector
$\mathbf{a}-\mathbf{b}=4\mathbf{i}-3\mathbf{j}-3\mathbf{k}$
This is the required resultant vector.

Answer

4\mathbf{i}-3\mathbf{j}-3\mathbf{k}

Question 3

2 markseasy

Given

\mathbf{a}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}

and

\mathbf{b}=3\mathbf{i}-\mathbf{j}

, find

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

Worked solution

Write down the given vectors
$\mathbf{a}=\mathbf{i}+2\mathbf{j}+2\mathbf{k},\quad \mathbf{b}=3\mathbf{i}-\mathbf{j}$
List the two vectors in component form.
Combine the corresponding components
$\left(2+3\right)\mathbf{i}+\left(4-1\right)\mathbf{j}+\left(4+0\right)\mathbf{k}=5\mathbf{i}+3\mathbf{j}+4\mathbf{k}$
Group the i, j and k components separately.
State the resultant vector
$2\mathbf{a}+\mathbf{b}=5\mathbf{i}+3\mathbf{j}+4\mathbf{k}$
This is the required resultant vector.

Answer

5\mathbf{i}+3\mathbf{j}+4\mathbf{k}

Question 4

2 markseasy

Given

\mathbf{a}=4\mathbf{i}+\mathbf{j}-\mathbf{k}

and

\mathbf{b}=2\mathbf{i}+2\mathbf{j}+3\mathbf{k}

, find

\mathbf{a}-\mathbf{b}

Worked solution

Write down the given vectors
$\mathbf{a}=4\mathbf{i}+\mathbf{j}-\mathbf{k},\quad \mathbf{b}=2\mathbf{i}+2\mathbf{j}+3\mathbf{k}$
List the two vectors in component form.
Combine the corresponding components
$\left(4-2\right)\mathbf{i}+\left(1-2\right)\mathbf{j}+\left(-1-3\right)\mathbf{k}=2\mathbf{i}-\mathbf{j}-4\mathbf{k}$
Group the i, j and k components separately.
State the resultant vector
$\mathbf{a}-\mathbf{b}=2\mathbf{i}-\mathbf{j}-4\mathbf{k}$
This is the required resultant vector.

Answer

2\mathbf{i}-\mathbf{j}-4\mathbf{k}

Question 5

2 markseasy

Find the magnitude of the vector

\mathbf{a}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}

Worked solution

Write the vector
$\mathbf{a}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}$
Identify the three components.
Apply the magnitude formula
$|\mathbf{a}|=\sqrt{\left(2\right)^{2}+\left(3\right)^{2}+\left(6\right)^{2}}$
Square each component and add them.
Simplify to state the magnitude
$|\mathbf{a}|=\sqrt{49}=7$
Evaluate the surd to give the magnitude.

Answer

7

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