Vectors

What you'll learn

How to write vectors using $i\mathbf{i}$ and $j\mathbf{j}$ components.
How to find a vector between two points and calculate its length.
How to add vectors to find resultants, especially forces.
How to find speed and bearing from a velocity vector.

1. What is a vector?

A scalar has size only, such as mass, time or temperature. A vector has size and direction, such as displacement, force or velocity.

In AS Maths, two-dimensional vectors are often written using $i\mathbf{i}$ and $j\mathbf{j}$ .

$i\mathbf{i}$ means one unit in the positive horizontal direction.
$j\mathbf{j}$ means one unit in the positive vertical direction.

So the vector $4i−3j4\mathbf{i}-3\mathbf{j}$ means “4 units right and 3 units down”.

$A component diagram showing the vector 4\mathbf{i}-3\mathbf{j} as 4 units right and 3 units down from the origin.$

Definition

Vector in component form

A vector written as $ai+bja\mathbf{i}+b\mathbf{j}$ has horizontal component $a$ and vertical component $b$ .

Key Idea

Components are instructions

Read $ai+bja\mathbf{i}+b\mathbf{j}$ as a movement: first $a$ units horizontally, then $b$ units vertically. A negative component means move in the opposite direction.

Example

Interpreting a vector

A particle starts at the point with position vector $2i+5j2\mathbf{i}+5\mathbf{j}$ . It moves through the vector $3i−7j3\mathbf{i}-7\mathbf{j}$ . Find its new position vector.

Start with the original position vector and add the movement vector:
$(2i+5j)+(3i−7j)(2\mathbf{i}+5\mathbf{j})+(3\mathbf{i}-7\mathbf{j})$
Add the $i\mathbf{i}$ components and the $j\mathbf{j}$ components separately:
$(2+3)i+(5−7)j=5i−2j\begin{aligned} (2+3)\mathbf{i}+(5-7)\mathbf{j} &=5\mathbf{i}-2\mathbf{j} \end{aligned}$
The new position vector is $5i−2j5\mathbf{i}-2\mathbf{j}$ .

Common Mistake

Mixing the components

Only combine like components: $i\mathbf{i}$ with $i\mathbf{i}$ , and $j\mathbf{j}$ with $j\mathbf{j}$ . Do not add the horizontal part to the vertical part.

2. Position vectors and vectors between points

A position vector tells you where a point is relative to the origin, usually called $O$ .

For example, if point $A$ has position vector $3i+4j3\mathbf{i}+4\mathbf{j}$ , then $A$ is 3 units right and 4 units up from the origin.

Definition

Position vector

The position vector of a point $A$ is the vector $OA⃗\vec{OA}$ , from the origin $O$ to the point $A$ .

To find the vector from $A$ to $B$ , subtract the position vector of $A$ from the position vector of $B$ :

AB⃗=OB⃗−OA⃗\vec{AB}=\vec{OB}-\vec{OA}

The diagram shows why: to travel from $A$ to $B$ , compare where $B$ is with where $A$ is.

$Position vectors 33m\vec{OA}0m and 33m\vec{OB}0m show that 33m\vec{AB}=\vec{OB}-\vec{OA}0m.$

Diagram showing position vectors OA, OB and displacement AB

Example

Finding a vector between two points

Point $A$ has position vector $4i−2j4\mathbf{i}-2\mathbf{j}$ and point $B$ has position vector $−3i+6j-3\mathbf{i}+6\mathbf{j}$ . Find $AB⃗\vec{AB}$ .

Write down the formula:
$AB⃗=OB⃗−OA⃗\vec{AB}=\vec{OB}-\vec{OA}$
Substitute the two position vectors:
$AB⃗=(−3i+6j)−(4i−2j)\vec{AB}=(-3\mathbf{i}+6\mathbf{j})-(4\mathbf{i}-2\mathbf{j})$
Subtract the components carefully:
$AB⃗=(−3−4)i+(6−(−2))j=−7i+8j\begin{aligned} \vec{AB} &=(-3-4)\mathbf{i}+(6-(-2))\mathbf{j}\\ &=-7\mathbf{i}+8\mathbf{j} \end{aligned}$
Therefore, $AB⃗=−7i+8j\vec{AB}=-7\mathbf{i}+8\mathbf{j}$ .

Tip

Order matters

$AB⃗\vec{AB}$ means “from $A$ to $B$ ”, so it is always final position minus starting position. Reversing the order gives $BA⃗\vec{BA}$ , which points the other way.

3. Magnitude: the length of a vector

The magnitude of a vector is its length. It is written using vertical bars, such as $∣v∣|\mathbf{v}|$ .

If $v=ai+bj\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ , then its magnitude comes from Pythagoras:

$The magnitude of a\mathbf{i}+b\mathbf{j} is the hypotenuse of a right triangle with sides a and b.$

∣v∣=a2+b2|\mathbf{v}|=\sqrt{a^2+b^2}

Definition

Magnitude

The magnitude of a vector is its length or size, ignoring direction.

Example

Finding the length of a vector

Find the length of the vector $−7i+8j-7\mathbf{i}+8\mathbf{j}$ .

Identify the horizontal and vertical components:
$a=−7,b=8a=-7,\qquad b=8$
Substitute into the magnitude formula:
$∣v∣=(−7)2+82|\mathbf{v}|=\sqrt{(-7)^2+8^2}$
Simplify:
$∣v∣=49+64=113\begin{aligned} |\mathbf{v}| &=\sqrt{49+64}\\ &=\sqrt{113} \end{aligned}$
So the length is $113\sqrt{113}$ , or about 10.6 to 3 significant figures.

Example

Using magnitude to find an unknown component

Given that $∣5i+kj∣=13|5\mathbf{i}+k\mathbf{j}|=13$ , find the possible values of $k$ .

Use the magnitude formula:
$∣5i+kj∣=52+k2|5\mathbf{i}+k\mathbf{j}|=\sqrt{5^2+k^2}$
Set it equal to 13:
$25+k2=13\sqrt{25+k^2}=13$
Square both sides:
$25+k^2=169$
Solve for $k$ :
$k2=144k=±12\begin{aligned} k^2&=144\\ k&=\pm 12 \end{aligned}$
The possible values are $k = 12$ or $k = - 12$ .

Common Mistake

Magnitude does not show direction

If you solve $k^2=144$ , there are two possible values unless the question gives extra information, such as $k > 0$ .

4. Adding vectors and resultant forces

When several vectors act together, you add their components. This is common with forces.

Multiple force vectors acting on a particle combine to give one resultant vector.

Definition

Resultant vector

The resultant vector is the single vector that has the same overall effect as two or more vectors acting together.

For example:

(2i+3j)+(5i−j)=7i+2j(2\mathbf{i}+3\mathbf{j})+(5\mathbf{i}-\mathbf{j})=7\mathbf{i}+2\mathbf{j}

Example

Finding a resultant force

Three forces act on a particle:

F1=−2i+6j,F2=7i−4j,F3=3i+5j\mathbf{F}_1=-2\mathbf{i}+6\mathbf{j},\qquad \mathbf{F}_2=7\mathbf{i}-4\mathbf{j},\qquad \mathbf{F}_3=3\mathbf{i}+5\mathbf{j}

Find the resultant force.

Add the three force vectors:
$R=F1+F2+F3\mathbf{R}=\mathbf{F}_1+\mathbf{F}_2+\mathbf{F}_3$
Substitute the vectors:
$R=(−2i+6j)+(7i−4j)+(3i+5j)\mathbf{R}=(-2\mathbf{i}+6\mathbf{j})+(7\mathbf{i}-4\mathbf{j})+(3\mathbf{i}+5\mathbf{j})$
Add the $i\mathbf{i}$ components and $j\mathbf{j}$ components separately:
$R=(−2+7+3)i+(6−4+5)j=8i+7j\begin{aligned} \mathbf{R} &=(-2+7+3)\mathbf{i}+(6-4+5)\mathbf{j}\\ &=8\mathbf{i}+7\mathbf{j} \end{aligned}$
The resultant force is $8i+7j8\mathbf{i}+7\mathbf{j}$ .

Example

Finding an unknown vector from a resultant

Vectors $a=−4i+9j\mathbf{a}=-4\mathbf{i}+9\mathbf{j}$ and $b=xi+yj\mathbf{b}=x\mathbf{i}+y\mathbf{j}$ have resultant $2i−5j2\mathbf{i}-5\mathbf{j}$ . Find $x$ and $y$ .

Write the vector equation:
$a+b=2i−5j\mathbf{a}+\mathbf{b}=2\mathbf{i}-5\mathbf{j}$
Substitute $a\mathbf{a}$ and $b\mathbf{b}$ :
$(−4i+9j)+(xi+yj)=2i−5j(-4\mathbf{i}+9\mathbf{j})+(x\mathbf{i}+y\mathbf{j})=2\mathbf{i}-5\mathbf{j}$
Compare the $i\mathbf{i}$ components:
$- 4 + x = 2$
Solve for $x$ :
$x = 6$
Compare the $j\mathbf{j}$ components:
$9 + y = - 5$
Solve for $y$ :
$y = - 14$

5. Vectors in triangles

In a triangle, vectors are often described using the vertices, such as $AB⃗\vec{AB}$ and $AC⃗\vec{AC}$ .

If you know two vectors starting from the same point, you can find the vector between their endpoints by subtracting.

For example:

BC⃗=AC⃗−AB⃗\vec{BC}=\vec{AC}-\vec{AB}

This works because travelling from $B$ to $C$ is the same as going from $B$ back to $A$ , then from $A$ to $C$ .

In triangle ABC, the vector from B to C can be found by going from B back to A and then from A to C.

Example

Finding a side vector in a triangle

In triangle $A B C$ ,

AB⃗=6i−2j,AC⃗=9i+4j\vec{AB}=6\mathbf{i}-2\mathbf{j},\qquad \vec{AC}=9\mathbf{i}+4\mathbf{j}

Find $BC⃗\vec{BC}$ and the length of $A B$ .

Use the relationship:
$BC⃗=AC⃗−AB⃗\vec{BC}=\vec{AC}-\vec{AB}$
Substitute the two known vectors:
$BC⃗=(9i+4j)−(6i−2j)\vec{BC}=(9\mathbf{i}+4\mathbf{j})-(6\mathbf{i}-2\mathbf{j})$
Subtract the components:
$BC⃗=(9−6)i+(4−(−2))j=3i+6j\begin{aligned} \vec{BC} &=(9-6)\mathbf{i}+(4-(-2))\mathbf{j}\\ &=3\mathbf{i}+6\mathbf{j} \end{aligned}$
To find the length of $A B$ , use the magnitude of $AB⃗=6i−2j\vec{AB}=6\mathbf{i}-2\mathbf{j}$ :
$∣AB⃗∣=62+(−2)2|\vec{AB}|=\sqrt{6^2+(-2)^2}$
Simplify:
$∣AB⃗∣=36+4=40=210\begin{aligned} |\vec{AB}| &=\sqrt{36+4}\\ &=\sqrt{40}\\ &=2\sqrt{10} \end{aligned}$

Common Mistake

Using the wrong direction in triangles

$BC⃗\vec{BC}$ is not usually $AB⃗−AC⃗\vec{AB}-\vec{AC}$ . Check the direction: from $B$ to $C$ means “position of $C$ minus position of $B$ ”.

6. Velocity vectors, speed and bearings

A velocity is a vector: it tells you both speed and direction. If $i\mathbf{i}$ is east and $j\mathbf{j}$ is north, then:

positive $i\mathbf{i}$ means east,
negative $i\mathbf{i}$ means west,
positive $j\mathbf{j}$ means north,
negative $j\mathbf{j}$ means south.

The speed is the magnitude of the velocity vector.

Definition

Bearing

A bearing is an angle measured clockwise from north. It is usually written as a three-figure angle, such as 037° or 126°.

This diagram shows a velocity vector pointing south-east, with the bearing measured clockwise from north.

A bearing is measured clockwise from north to the direction of travel.

Bearing diagram for a velocity vector with east and south components

Example

Finding speed and bearing from velocity

A boat has velocity $s−1(6\mathbf{i}-8\mathbf{j})\text{ m s}^{-1}$ , where $i\mathbf{i}$ points east and $j\mathbf{j}$ points north. Find its speed and bearing.

The boat’s velocity forms a right triangle with east component 6 and south component 8, so the bearing is measured clockwise from north.

The speed is the magnitude of the velocity vector:
$speed=62+(−8)2\text{speed}=\sqrt{6^2+(-8)^2}$
Calculate the speed:
$speed=36+64=10\begin{aligned} \text{speed} &=\sqrt{36+64}\\ &=10 \end{aligned}$
The vector points 6 units east and 8 units south, so it is in the south-east quadrant.
Find the angle below east:
$θ=tan⁡−1(86)≈53.1∘\theta=\tan^{-1}\left(\frac{8}{6}\right)\approx 53.1^\circ$
Bearings are measured clockwise from north. East is 090°, so add the angle below east:
$90∘+53.1∘=143.1∘90^\circ+53.1^\circ=143.1^\circ$
The bearing is approximately 143° to the nearest degree.

Tip

Bearing sanity check

Before calculating, decide the quadrant. A vector with positive $i\mathbf{i}$ and negative $j\mathbf{j}$ points south-east, so its bearing must be between 090° and 180°.

Exam technique

In the exam

Write vectors in component form and keep the $i\mathbf{i}$ and $j\mathbf{j}$ parts separate throughout.
For $AB⃗\vec{AB}$ , always use final minus initial: $OB⃗−OA⃗\vec{OB}-\vec{OA}$ .
For lengths, use Pythagoras: square both components, add, then square root.
For bearings, sketch a quick compass diagram first so you know whether to add or subtract your angle.

Self review

Check yourself

If $A$ has position vector $2i−3j2\mathbf{i}-3\mathbf{j}$ and $B$ has position vector $7i+4j7\mathbf{i}+4\mathbf{j}$ , how would you find $AB⃗\vec{AB}$ ?
Why can $k^2=25$ give two possible values of $k$ ?
A velocity vector has positive $i\mathbf{i}$ component and negative $j\mathbf{j}$ component. Which quadrant is it travelling in?

Vectors

What you'll learn

How to write vectors using $i\mathbf{i}$ and $j\mathbf{j}$ components.
How to find a vector between two points and calculate its length.
How to add vectors to find resultants, especially forces.
How to find speed and bearing from a velocity vector.

1. What is a vector?

A scalar has size only, such as mass, time or temperature. A vector has size and direction, such as displacement, force or velocity.

In AS Maths, two-dimensional vectors are often written using $i\mathbf{i}$ and $j\mathbf{j}$ .

$i\mathbf{i}$ means one unit in the positive horizontal direction.
$j\mathbf{j}$ means one unit in the positive vertical direction.

So the vector $4i−3j4\mathbf{i}-3\mathbf{j}$ means “4 units right and 3 units down”.

$A component diagram showing the vector 4\mathbf{i}-3\mathbf{j} as 4 units right and 3 units down from the origin.$

Definition

Vector in component form

A vector written as $ai+bja\mathbf{i}+b\mathbf{j}$ has horizontal component $a$ and vertical component $b$ .

Key Idea

Components are instructions

Read $ai+bja\mathbf{i}+b\mathbf{j}$ as a movement: first $a$ units horizontally, then $b$ units vertically. A negative component means move in the opposite direction.

Example

Interpreting a vector

A particle starts at the point with position vector $2i+5j2\mathbf{i}+5\mathbf{j}$ . It moves through the vector $3i−7j3\mathbf{i}-7\mathbf{j}$ . Find its new position vector.

Start with the original position vector and add the movement vector:
$(2i+5j)+(3i−7j)(2\mathbf{i}+5\mathbf{j})+(3\mathbf{i}-7\mathbf{j})$
Add the $i\mathbf{i}$ components and the $j\mathbf{j}$ components separately:
$(2+3)i+(5−7)j=5i−2j\begin{aligned} (2+3)\mathbf{i}+(5-7)\mathbf{j} &=5\mathbf{i}-2\mathbf{j} \end{aligned}$
The new position vector is $5i−2j5\mathbf{i}-2\mathbf{j}$ .

Common Mistake

Mixing the components

Only combine like components: $i\mathbf{i}$ with $i\mathbf{i}$ , and $j\mathbf{j}$ with $j\mathbf{j}$ . Do not add the horizontal part to the vertical part.

2. Position vectors and vectors between points

A position vector tells you where a point is relative to the origin, usually called $O$ .

For example, if point $A$ has position vector $3i+4j3\mathbf{i}+4\mathbf{j}$ , then $A$ is 3 units right and 4 units up from the origin.

Definition

Position vector

The position vector of a point $A$ is the vector $OA⃗\vec{OA}$ , from the origin $O$ to the point $A$ .

To find the vector from $A$ to $B$ , subtract the position vector of $A$ from the position vector of $B$ :

AB⃗=OB⃗−OA⃗\vec{AB}=\vec{OB}-\vec{OA}

The diagram shows why: to travel from $A$ to $B$ , compare where $B$ is with where $A$ is.

$Position vectors 33m\vec{OA}0m and 33m\vec{OB}0m show that 33m\vec{AB}=\vec{OB}-\vec{OA}0m.$

Diagram showing position vectors OA, OB and displacement AB

Example

Finding a vector between two points

Point $A$ has position vector $4i−2j4\mathbf{i}-2\mathbf{j}$ and point $B$ has position vector $−3i+6j-3\mathbf{i}+6\mathbf{j}$ . Find $AB⃗\vec{AB}$ .

Write down the formula:
$AB⃗=OB⃗−OA⃗\vec{AB}=\vec{OB}-\vec{OA}$
Substitute the two position vectors:
$AB⃗=(−3i+6j)−(4i−2j)\vec{AB}=(-3\mathbf{i}+6\mathbf{j})-(4\mathbf{i}-2\mathbf{j})$
Subtract the components carefully:
$AB⃗=(−3−4)i+(6−(−2))j=−7i+8j\begin{aligned} \vec{AB} &=(-3-4)\mathbf{i}+(6-(-2))\mathbf{j}\\ &=-7\mathbf{i}+8\mathbf{j} \end{aligned}$
Therefore, $AB⃗=−7i+8j\vec{AB}=-7\mathbf{i}+8\mathbf{j}$ .

Tip

Order matters

$AB⃗\vec{AB}$ means “from $A$ to $B$ ”, so it is always final position minus starting position. Reversing the order gives $BA⃗\vec{BA}$ , which points the other way.

3. Magnitude: the length of a vector

The magnitude of a vector is its length. It is written using vertical bars, such as $∣v∣|\mathbf{v}|$ .

If $v=ai+bj\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ , then its magnitude comes from Pythagoras:

$The magnitude of a\mathbf{i}+b\mathbf{j} is the hypotenuse of a right triangle with sides a and b.$

∣v∣=a2+b2|\mathbf{v}|=\sqrt{a^2+b^2}

Definition

Magnitude

The magnitude of a vector is its length or size, ignoring direction.

Example

Finding the length of a vector

Find the length of the vector $−7i+8j-7\mathbf{i}+8\mathbf{j}$ .

Identify the horizontal and vertical components:
$a=−7,b=8a=-7,\qquad b=8$
Substitute into the magnitude formula:
$∣v∣=(−7)2+82|\mathbf{v}|=\sqrt{(-7)^2+8^2}$
Simplify:
$∣v∣=49+64=113\begin{aligned} |\mathbf{v}| &=\sqrt{49+64}\\ &=\sqrt{113} \end{aligned}$
So the length is $113\sqrt{113}$ , or about 10.6 to 3 significant figures.

Example

Using magnitude to find an unknown component

Given that $∣5i+kj∣=13|5\mathbf{i}+k\mathbf{j}|=13$ , find the possible values of $k$ .

Use the magnitude formula:
$∣5i+kj∣=52+k2|5\mathbf{i}+k\mathbf{j}|=\sqrt{5^2+k^2}$
Set it equal to 13:
$25+k2=13\sqrt{25+k^2}=13$
Square both sides:
$25+k^2=169$
Solve for $k$ :
$k2=144k=±12\begin{aligned} k^2&=144\\ k&=\pm 12 \end{aligned}$
The possible values are $k = 12$ or $k = - 12$ .

Common Mistake

Magnitude does not show direction

If you solve $k^2=144$ , there are two possible values unless the question gives extra information, such as $k > 0$ .

4. Adding vectors and resultant forces

When several vectors act together, you add their components. This is common with forces.

Multiple force vectors acting on a particle combine to give one resultant vector.

Definition

Resultant vector

The resultant vector is the single vector that has the same overall effect as two or more vectors acting together.

For example:

(2i+3j)+(5i−j)=7i+2j(2\mathbf{i}+3\mathbf{j})+(5\mathbf{i}-\mathbf{j})=7\mathbf{i}+2\mathbf{j}

Example

Finding a resultant force

Three forces act on a particle:

F1=−2i+6j,F2=7i−4j,F3=3i+5j\mathbf{F}_1=-2\mathbf{i}+6\mathbf{j},\qquad \mathbf{F}_2=7\mathbf{i}-4\mathbf{j},\qquad \mathbf{F}_3=3\mathbf{i}+5\mathbf{j}

Find the resultant force.

Add the three force vectors:
$R=F1+F2+F3\mathbf{R}=\mathbf{F}_1+\mathbf{F}_2+\mathbf{F}_3$
Substitute the vectors:
$R=(−2i+6j)+(7i−4j)+(3i+5j)\mathbf{R}=(-2\mathbf{i}+6\mathbf{j})+(7\mathbf{i}-4\mathbf{j})+(3\mathbf{i}+5\mathbf{j})$
Add the $i\mathbf{i}$ components and $j\mathbf{j}$ components separately:
$R=(−2+7+3)i+(6−4+5)j=8i+7j\begin{aligned} \mathbf{R} &=(-2+7+3)\mathbf{i}+(6-4+5)\mathbf{j}\\ &=8\mathbf{i}+7\mathbf{j} \end{aligned}$
The resultant force is $8i+7j8\mathbf{i}+7\mathbf{j}$ .

Example

Finding an unknown vector from a resultant

Vectors $a=−4i+9j\mathbf{a}=-4\mathbf{i}+9\mathbf{j}$ and $b=xi+yj\mathbf{b}=x\mathbf{i}+y\mathbf{j}$ have resultant $2i−5j2\mathbf{i}-5\mathbf{j}$ . Find $x$ and $y$ .

Write the vector equation:
$a+b=2i−5j\mathbf{a}+\mathbf{b}=2\mathbf{i}-5\mathbf{j}$
Substitute $a\mathbf{a}$ and $b\mathbf{b}$ :
$(−4i+9j)+(xi+yj)=2i−5j(-4\mathbf{i}+9\mathbf{j})+(x\mathbf{i}+y\mathbf{j})=2\mathbf{i}-5\mathbf{j}$
Compare the $i\mathbf{i}$ components:
$- 4 + x = 2$
Solve for $x$ :
$x = 6$
Compare the $j\mathbf{j}$ components:
$9 + y = - 5$
Solve for $y$ :
$y = - 14$

5. Vectors in triangles

In a triangle, vectors are often described using the vertices, such as $AB⃗\vec{AB}$ and $AC⃗\vec{AC}$ .

If you know two vectors starting from the same point, you can find the vector between their endpoints by subtracting.

For example:

BC⃗=AC⃗−AB⃗\vec{BC}=\vec{AC}-\vec{AB}

This works because travelling from $B$ to $C$ is the same as going from $B$ back to $A$ , then from $A$ to $C$ .

In triangle ABC, the vector from B to C can be found by going from B back to A and then from A to C.

Example

Finding a side vector in a triangle

In triangle $A B C$ ,

AB⃗=6i−2j,AC⃗=9i+4j\vec{AB}=6\mathbf{i}-2\mathbf{j},\qquad \vec{AC}=9\mathbf{i}+4\mathbf{j}

Find $BC⃗\vec{BC}$ and the length of $A B$ .

Use the relationship:
$BC⃗=AC⃗−AB⃗\vec{BC}=\vec{AC}-\vec{AB}$
Substitute the two known vectors:
$BC⃗=(9i+4j)−(6i−2j)\vec{BC}=(9\mathbf{i}+4\mathbf{j})-(6\mathbf{i}-2\mathbf{j})$
Subtract the components:
$BC⃗=(9−6)i+(4−(−2))j=3i+6j\begin{aligned} \vec{BC} &=(9-6)\mathbf{i}+(4-(-2))\mathbf{j}\\ &=3\mathbf{i}+6\mathbf{j} \end{aligned}$
To find the length of $A B$ , use the magnitude of $AB⃗=6i−2j\vec{AB}=6\mathbf{i}-2\mathbf{j}$ :
$∣AB⃗∣=62+(−2)2|\vec{AB}|=\sqrt{6^2+(-2)^2}$
Simplify:
$∣AB⃗∣=36+4=40=210\begin{aligned} |\vec{AB}| &=\sqrt{36+4}\\ &=\sqrt{40}\\ &=2\sqrt{10} \end{aligned}$

Common Mistake

Using the wrong direction in triangles

$BC⃗\vec{BC}$ is not usually $AB⃗−AC⃗\vec{AB}-\vec{AC}$ . Check the direction: from $B$ to $C$ means “position of $C$ minus position of $B$ ”.

6. Velocity vectors, speed and bearings

A velocity is a vector: it tells you both speed and direction. If $i\mathbf{i}$ is east and $j\mathbf{j}$ is north, then:

positive $i\mathbf{i}$ means east,
negative $i\mathbf{i}$ means west,
positive $j\mathbf{j}$ means north,
negative $j\mathbf{j}$ means south.

The speed is the magnitude of the velocity vector.

Definition

Bearing

A bearing is an angle measured clockwise from north. It is usually written as a three-figure angle, such as 037° or 126°.

This diagram shows a velocity vector pointing south-east, with the bearing measured clockwise from north.

A bearing is measured clockwise from north to the direction of travel.

Bearing diagram for a velocity vector with east and south components

Example

Finding speed and bearing from velocity

A boat has velocity $s−1(6\mathbf{i}-8\mathbf{j})\text{ m s}^{-1}$ , where $i\mathbf{i}$ points east and $j\mathbf{j}$ points north. Find its speed and bearing.

The boat’s velocity forms a right triangle with east component 6 and south component 8, so the bearing is measured clockwise from north.

The speed is the magnitude of the velocity vector:
$speed=62+(−8)2\text{speed}=\sqrt{6^2+(-8)^2}$
Calculate the speed:
$speed=36+64=10\begin{aligned} \text{speed} &=\sqrt{36+64}\\ &=10 \end{aligned}$
The vector points 6 units east and 8 units south, so it is in the south-east quadrant.
Find the angle below east:
$θ=tan⁡−1(86)≈53.1∘\theta=\tan^{-1}\left(\frac{8}{6}\right)\approx 53.1^\circ$
Bearings are measured clockwise from north. East is 090°, so add the angle below east:
$90∘+53.1∘=143.1∘90^\circ+53.1^\circ=143.1^\circ$
The bearing is approximately 143° to the nearest degree.

Tip

Bearing sanity check

Before calculating, decide the quadrant. A vector with positive $i\mathbf{i}$ and negative $j\mathbf{j}$ points south-east, so its bearing must be between 090° and 180°.

Exam technique

In the exam

Write vectors in component form and keep the $i\mathbf{i}$ and $j\mathbf{j}$ parts separate throughout.
For $AB⃗\vec{AB}$ , always use final minus initial: $OB⃗−OA⃗\vec{OB}-\vec{OA}$ .
For lengths, use Pythagoras: square both components, add, then square root.
For bearings, sketch a quick compass diagram first so you know whether to add or subtract your angle.

Self review

Check yourself

If $A$ has position vector $2i−3j2\mathbf{i}-3\mathbf{j}$ and $B$ has position vector $7i+4j7\mathbf{i}+4\mathbf{j}$ , how would you find $AB⃗\vec{AB}$ ?
Why can $k^2=25$ give two possible values of $k$ ?
A velocity vector has positive $i\mathbf{i}$ component and negative $j\mathbf{j}$ component. Which quadrant is it travelling in?

Vectors

What you'll learn

1. What is a vector?

2. Position vectors and vectors between points

3. Magnitude: the length of a vector

4. Adding vectors and resultant forces

5. Vectors in triangles

6. Velocity vectors, speed and bearings

Recap questions

You've reached the end

Vectors

What you'll learn

1. What is a vector?

2. Position vectors and vectors between points

3. Magnitude: the length of a vector

4. Adding vectors and resultant forces

5. Vectors in triangles

6. Velocity vectors, speed and bearings

Recap questions

You've reached the end

Vectors

What you'll learn

1. What is a vector?

2. Position vectors and vectors between points

3. Magnitude: the length of a vector

4. Adding vectors and resultant forces

5. Vectors in triangles

6. Velocity vectors, speed and bearings

Recap questions

You've reached the end

Vectors Revision Guide

Vectors

What you'll learn

1. What is a vector?

2. Position vectors and vectors between points

3. Magnitude: the length of a vector

4. Adding vectors and resultant forces

5. Vectors in triangles

6. Velocity vectors, speed and bearings

Recap questions

You've reached the end

Vectors Revision Guide