1.10.4 Basic operations on vectors

What you'll learn

How to add vectors using diagrams and components.
How multiplying a vector by a scalar changes its length and direction.
How vector subtraction follows from adding a negative vector.
How to interpret vector calculations geometrically and check that your answers are sensible.

Starting Point: What Is a Vector?

A vector is a quantity with both magnitude and direction. Its magnitude is its size or length.

For example, a displacement of 5 m east is a vector because it specifies both a distance and a direction. By contrast, a distance of 5 m is a scalar, because it has magnitude but no direction.

Vectors are often written in bold, such as $a\mathbf{a}$ , or as a column vector:

a=(32).\mathbf{a}= \begin{pmatrix} 3\\ 2 \end{pmatrix}.

The top component gives the movement in the horizontal direction, while the bottom component gives the movement in the vertical direction. Therefore, this vector represents a movement of 3 units to the right and 2 units upwards.

A negative component reverses the movement along that axis. For example,

(−41)\begin{pmatrix} -4\\ 1 \end{pmatrix}

represents 4 units to the left and 1 unit upwards.

Definition

Equal vectors

Two vectors are equal if they have the same magnitude and the same direction. They do not need to begin at the same point.

This means that you may slide a vector parallel to itself without changing the vector, provided you do not rotate it or change its length. This is important when adding vectors diagrammatically.

Adding Vectors Diagrammatically

Vectors describe movements, so adding vectors means carrying out one movement followed by another.

To construct $a+b\mathbf{a}+\mathbf{b}$ :

Draw $a\mathbf{a}$ .
Place the tail of $b\mathbf{b}$ at the head of $a\mathbf{a}$ .
Draw the resultant vector from the tail of $a\mathbf{a}$ to the head of $b\mathbf{b}$ .

This is called the head-to-tail rule. The direct vector is known as the resultant, meaning the single vector with the same overall effect as the two original vectors.

The same addition can also be shown using the parallelogram rule. Draw $a\mathbf{a}$ and $b\mathbf{b}$ from the same starting point, complete a parallelogram, and draw its diagonal from the common starting point. That diagonal is $a+b\mathbf{a}+\mathbf{b}$ .

Three vector diagrams showing head-to-tail addition, parallelogram addition and the geometric effects of multiplying a vector by 2 and by negative 1

Example

Adding two displacements

A particle moves along vector $a\mathbf{a}$ and then along vector $b\mathbf{b}$ . Describe how to construct its resultant displacement.

Draw $a\mathbf{a}$ with its arrow showing the direction of the first movement.
Translate $b\mathbf{b}$ without rotating or resizing it, placing its tail at the head of $a\mathbf{a}$ .
Join the starting point of $a\mathbf{a}$ to the finishing point of $b\mathbf{b}$ . The resulting arrow is $a+b\mathbf{a}+\mathbf{b}$ .

Common Mistake

Joining the wrong points

For head-to-tail addition, the resultant runs from the start of the first vector to the end of the second vector. Reversing these points gives $−(a+b)-(\mathbf{a}+\mathbf{b})$ .

Adding Vectors Using Components

To add vectors algebraically, add corresponding components:

(ab)+(cd)=(a+cb+d).\begin{pmatrix} a\\ b \end{pmatrix} + \begin{pmatrix} c\\ d \end{pmatrix} = \begin{pmatrix} a+c\\ b+d \end{pmatrix}.

You add the horizontal components together and independently add the vertical components together.

Example

Adding column vectors

Given

a=(4−3),b=(−15),\mathbf{a}= \begin{pmatrix} 4\\ -3 \end{pmatrix}, \qquad \mathbf{b}= \begin{pmatrix} -1\\ 5 \end{pmatrix},

find $a+b\mathbf{a}+\mathbf{b}$ .

Add the horizontal components: $4 + (- 1) = 3$ .
Add the vertical components: $- 3 + 5 = 2$ .
Combine the results in the original component order:

a+b=(4+(−1)−3+5)=(32).\mathbf{a}+\mathbf{b} = \begin{pmatrix} 4+(-1)\\ -3+5 \end{pmatrix} = \begin{pmatrix} 3\\ 2 \end{pmatrix}.

Vector addition is commutative, so changing the order does not change the resultant:

a+b=b+a.\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}.

Geometrically, either order takes you to the same final point. The two possible routes form adjacent sides of a parallelogram.

Vector addition is also associative:

(a+b)+c=a+(b+c).(\mathbf{a}+\mathbf{b})+\mathbf{c} = \mathbf{a}+(\mathbf{b}+\mathbf{c}).

This means that when adding three or more vectors, the way in which you group them does not affect the result.

The Zero Vector and Negative Vectors

The zero vector has magnitude zero and no particular direction. In two dimensions it is written as

0=(00).\mathbf{0}= \begin{pmatrix} 0\\ 0 \end{pmatrix}.

Adding the zero vector has no effect:

a+0=a.\mathbf{a}+\mathbf{0}=\mathbf{a}.

The negative of $a\mathbf{a}$ , written $−a-\mathbf{a}$ , has the same magnitude as $a\mathbf{a}$ but points in the opposite direction. Component by component,

−(ab)=(−a−b).-\begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} -a\\ -b \end{pmatrix}.

A vector and its negative cancel:

a+(−a)=0.\mathbf{a}+(-\mathbf{a})=\mathbf{0}.

Key Idea

Reversing a vector

If $AB→=a\overrightarrow{AB}=\mathbf{a}$ , then reversing the start and end points gives $BA→=−a\overrightarrow{BA}=-\mathbf{a}$ .

Subtracting Vectors

Subtracting a vector means adding its negative:

a−b=a+(−b).\mathbf{a}-\mathbf{b} = \mathbf{a}+(-\mathbf{b}).

Using components, subtract corresponding entries:

(ab)−(cd)=(a−cb−d).\begin{pmatrix} a\\ b \end{pmatrix} - \begin{pmatrix} c\\ d \end{pmatrix} = \begin{pmatrix} a-c\\ b-d \end{pmatrix}.

Geometrically, if $a\mathbf{a}$ and $b\mathbf{b}$ are drawn from the same starting point, then $a−b\mathbf{a}-\mathbf{b}$ points from the head of $b\mathbf{b}$ to the head of $a\mathbf{a}$ .

Example

Finding a vector between two points

From a common origin $O$ ,

OA→=(61),OB→=(2−3).\overrightarrow{OA}= \begin{pmatrix} 6\\ 1 \end{pmatrix}, \qquad \overrightarrow{OB}= \begin{pmatrix} 2\\ -3 \end{pmatrix}.

Find $AB→\overrightarrow{AB}$ .

Travelling from $A$ to $B$ can be viewed as travelling from $A$ back to $O$ , then from $O$ to $B$ :

AB→=AO→+OB→=−OA→+OB→.\overrightarrow{AB} = \overrightarrow{AO}+\overrightarrow{OB} = -\overrightarrow{OA}+\overrightarrow{OB}.

Therefore, subtract the position vector of the starting point from that of the finishing point:

AB→=OB→−OA→.\overrightarrow{AB} = \overrightarrow{OB}-\overrightarrow{OA}.

Subtract corresponding components:

AB→=(2−3)−(61)=(−4−4).\overrightarrow{AB} = \begin{pmatrix} 2\\ -3 \end{pmatrix} - \begin{pmatrix} 6\\ 1 \end{pmatrix} = \begin{pmatrix} -4\\ -4 \end{pmatrix}.

Tip

Vector from one point to another

For a vector from $A$ to $B$ , use finish minus start: $AB→=OB→−OA→\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$ .

Multiplying a Vector by a Scalar

A scalar multiple of a vector is produced by multiplying the vector by an ordinary number, called a scalar.

For a scalar $k$ ,

\begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} ka\\ kb \end{pmatrix}.

Geometrically:

If $k > 0$ , then $kak\mathbf{a}$ points in the same direction as $a\mathbf{a}$ .
If $k < 0$ , then $kak\mathbf{a}$ points in the opposite direction.
Its magnitude is multiplied by $∣ k ∣$ .
If $k = 0$ , the result is the zero vector.

Therefore, $2a2\mathbf{a}$ is twice as long as $a\mathbf{a}$ in the same direction, while $−12a-\frac{1}{2}\mathbf{a}$ is half as long and points in the opposite direction.

Example

Combining scalar multiples

Given

a=(3−2),b=(−14),\mathbf{a}= \begin{pmatrix} 3\\ -2 \end{pmatrix}, \qquad \mathbf{b}= \begin{pmatrix} -1\\ 4 \end{pmatrix},

find $2a−3b2\mathbf{a}-3\mathbf{b}$ .

Multiply every component of $a\mathbf{a}$ by 2:

2a=(6−4).2\mathbf{a} = \begin{pmatrix} 6\\ -4 \end{pmatrix}.

Multiply every component of $b\mathbf{b}$ by 3:

3b=(−312).3\mathbf{b} = \begin{pmatrix} -3\\ 12 \end{pmatrix}.

Subtract corresponding components:

2a−3b=(6−4)−(−312)=(9−16).2\mathbf{a}-3\mathbf{b} = \begin{pmatrix} 6\\ -4 \end{pmatrix} - \begin{pmatrix} -3\\ 12 \end{pmatrix} = \begin{pmatrix} 9\\ -16 \end{pmatrix}.

Common Mistake

Scaling only one component

A scalar multiplies every component of the vector. For example, $3(ab)=(3a3b)3\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}3a\\3b\end{pmatrix}$ .

Scalar multiplication distributes over vector addition:

k(a+b)=ka+kb.k(\mathbf{a}+\mathbf{b})=k\mathbf{a}+k\mathbf{b}.

It also satisfies

(k+m)a=ka+ma.(k+m)\mathbf{a}=k\mathbf{a}+m\mathbf{a}.

These rules allow you to expand and simplify vector expressions in much the same way as ordinary algebra.

Parallel Vectors

Non-zero vectors are parallel when one is a scalar multiple of the other:

b=ka\mathbf{b}=k\mathbf{a}

for some scalar $k$ .

If $k$ is positive, the vectors point in the same direction. If $k$ is negative, they point in opposite directions.

Example

Checking whether vectors are parallel

Determine whether

p=(−69),q=(2−3)\mathbf{p}= \begin{pmatrix} -6\\ 9 \end{pmatrix}, \qquad \mathbf{q}= \begin{pmatrix} 2\\ -3 \end{pmatrix}

are parallel.

Compare the first components: to change 2 into negative 6, multiply by negative 3.
Check the same scalar against the second components: $−3×(−3)=9-3\times(-3)=9$ .
The same scalar works for both components, so $p=−3q\mathbf{p}=-3\mathbf{q}$ . The vectors are parallel and point in opposite directions.

Exam technique

In the exam

Keep components vertically aligned, and operate on corresponding components only.
For a vector between two points, use finish minus start and check the arrow direction.
Use the diagram as a sense check: a negative scalar reverses direction, while vector addition should represent consecutive movements.
Keep answers exact unless the question specifically asks for a decimal.

Self review

Check yourself

How would you construct $a+b\mathbf{a}+\mathbf{b}$ using the head-to-tail rule?
Given $a=(2−5)\mathbf{a}=\begin{pmatrix}2\\-5\end{pmatrix}$ and $b=(−31)\mathbf{b}=\begin{pmatrix}-3\\1\end{pmatrix}$ , can you find $3a−2b3\mathbf{a}-2\mathbf{b}$ ?
What does a negative scalar do to the magnitude and direction of a vector?

What you'll learn

How to add vectors using diagrams and components.
How multiplying a vector by a scalar changes its length and direction.
How vector subtraction follows from adding a negative vector.
How to interpret vector calculations geometrically and check that your answers are sensible.

Starting Point: What Is a Vector?

A vector is a quantity with both magnitude and direction. Its magnitude is its size or length.

For example, a displacement of 5 m east is a vector because it specifies both a distance and a direction. By contrast, a distance of 5 m is a scalar, because it has magnitude but no direction.

Vectors are often written in bold, such as $a\mathbf{a}$ , or as a column vector:

a=(32).\mathbf{a}= \begin{pmatrix} 3\\ 2 \end{pmatrix}.

A negative component reverses the movement along that axis. For example,

(−41)\begin{pmatrix} -4\\ 1 \end{pmatrix}

represents 4 units to the left and 1 unit upwards.

Definition

Equal vectors

Two vectors are equal if they have the same magnitude and the same direction. They do not need to begin at the same point.

This means that you may slide a vector parallel to itself without changing the vector, provided you do not rotate it or change its length. This is important when adding vectors diagrammatically.

Adding Vectors Diagrammatically

Vectors describe movements, so adding vectors means carrying out one movement followed by another.

To construct $a+b\mathbf{a}+\mathbf{b}$ :

Draw $a\mathbf{a}$ .
Place the tail of $b\mathbf{b}$ at the head of $a\mathbf{a}$ .
Draw the resultant vector from the tail of $a\mathbf{a}$ to the head of $b\mathbf{b}$ .

This is called the head-to-tail rule. The direct vector is known as the resultant, meaning the single vector with the same overall effect as the two original vectors.

Three vector diagrams showing head-to-tail addition, parallelogram addition and the geometric effects of multiplying a vector by 2 and by negative 1

Example

Adding two displacements

A particle moves along vector $a\mathbf{a}$ and then along vector $b\mathbf{b}$ . Describe how to construct its resultant displacement.

Draw $a\mathbf{a}$ with its arrow showing the direction of the first movement.
Translate $b\mathbf{b}$ without rotating or resizing it, placing its tail at the head of $a\mathbf{a}$ .
Join the starting point of $a\mathbf{a}$ to the finishing point of $b\mathbf{b}$ . The resulting arrow is $a+b\mathbf{a}+\mathbf{b}$ .

Common Mistake

Joining the wrong points

For head-to-tail addition, the resultant runs from the start of the first vector to the end of the second vector. Reversing these points gives $−(a+b)-(\mathbf{a}+\mathbf{b})$ .

Adding Vectors Using Components

To add vectors algebraically, add corresponding components:

(ab)+(cd)=(a+cb+d).\begin{pmatrix} a\\ b \end{pmatrix} + \begin{pmatrix} c\\ d \end{pmatrix} = \begin{pmatrix} a+c\\ b+d \end{pmatrix}.

You add the horizontal components together and independently add the vertical components together.

Example

Adding column vectors

Given

a=(4−3),b=(−15),\mathbf{a}= \begin{pmatrix} 4\\ -3 \end{pmatrix}, \qquad \mathbf{b}= \begin{pmatrix} -1\\ 5 \end{pmatrix},

find $a+b\mathbf{a}+\mathbf{b}$ .

Add the horizontal components: $4 + (- 1) = 3$ .
Add the vertical components: $- 3 + 5 = 2$ .
Combine the results in the original component order:

a+b=(4+(−1)−3+5)=(32).\mathbf{a}+\mathbf{b} = \begin{pmatrix} 4+(-1)\\ -3+5 \end{pmatrix} = \begin{pmatrix} 3\\ 2 \end{pmatrix}.

Vector addition is commutative, so changing the order does not change the resultant:

a+b=b+a.\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}.

Geometrically, either order takes you to the same final point. The two possible routes form adjacent sides of a parallelogram.

Vector addition is also associative:

(a+b)+c=a+(b+c).(\mathbf{a}+\mathbf{b})+\mathbf{c} = \mathbf{a}+(\mathbf{b}+\mathbf{c}).

This means that when adding three or more vectors, the way in which you group them does not affect the result.

The Zero Vector and Negative Vectors

The zero vector has magnitude zero and no particular direction. In two dimensions it is written as

0=(00).\mathbf{0}= \begin{pmatrix} 0\\ 0 \end{pmatrix}.

Adding the zero vector has no effect:

a+0=a.\mathbf{a}+\mathbf{0}=\mathbf{a}.

The negative of $a\mathbf{a}$ , written $−a-\mathbf{a}$ , has the same magnitude as $a\mathbf{a}$ but points in the opposite direction. Component by component,

−(ab)=(−a−b).-\begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} -a\\ -b \end{pmatrix}.

A vector and its negative cancel:

a+(−a)=0.\mathbf{a}+(-\mathbf{a})=\mathbf{0}.

Key Idea

Reversing a vector

If $AB→=a\overrightarrow{AB}=\mathbf{a}$ , then reversing the start and end points gives $BA→=−a\overrightarrow{BA}=-\mathbf{a}$ .

Subtracting Vectors

Subtracting a vector means adding its negative:

a−b=a+(−b).\mathbf{a}-\mathbf{b} = \mathbf{a}+(-\mathbf{b}).

Using components, subtract corresponding entries:

(ab)−(cd)=(a−cb−d).\begin{pmatrix} a\\ b \end{pmatrix} - \begin{pmatrix} c\\ d \end{pmatrix} = \begin{pmatrix} a-c\\ b-d \end{pmatrix}.

Geometrically, if $a\mathbf{a}$ and $b\mathbf{b}$ are drawn from the same starting point, then $a−b\mathbf{a}-\mathbf{b}$ points from the head of $b\mathbf{b}$ to the head of $a\mathbf{a}$ .

Example

Finding a vector between two points

From a common origin $O$ ,

OA→=(61),OB→=(2−3).\overrightarrow{OA}= \begin{pmatrix} 6\\ 1 \end{pmatrix}, \qquad \overrightarrow{OB}= \begin{pmatrix} 2\\ -3 \end{pmatrix}.

Find $AB→\overrightarrow{AB}$ .

Travelling from $A$ to $B$ can be viewed as travelling from $A$ back to $O$ , then from $O$ to $B$ :

AB→=AO→+OB→=−OA→+OB→.\overrightarrow{AB} = \overrightarrow{AO}+\overrightarrow{OB} = -\overrightarrow{OA}+\overrightarrow{OB}.

Therefore, subtract the position vector of the starting point from that of the finishing point:

AB→=OB→−OA→.\overrightarrow{AB} = \overrightarrow{OB}-\overrightarrow{OA}.

Subtract corresponding components:

AB→=(2−3)−(61)=(−4−4).\overrightarrow{AB} = \begin{pmatrix} 2\\ -3 \end{pmatrix} - \begin{pmatrix} 6\\ 1 \end{pmatrix} = \begin{pmatrix} -4\\ -4 \end{pmatrix}.

Tip

Vector from one point to another

For a vector from $A$ to $B$ , use finish minus start: $AB→=OB→−OA→\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$ .

Multiplying a Vector by a Scalar

A scalar multiple of a vector is produced by multiplying the vector by an ordinary number, called a scalar.

For a scalar $k$ ,

\begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} ka\\ kb \end{pmatrix}.

Geometrically:

If $k > 0$ , then $kak\mathbf{a}$ points in the same direction as $a\mathbf{a}$ .
If $k < 0$ , then $kak\mathbf{a}$ points in the opposite direction.
Its magnitude is multiplied by $∣ k ∣$ .
If $k = 0$ , the result is the zero vector.

Therefore, $2a2\mathbf{a}$ is twice as long as $a\mathbf{a}$ in the same direction, while $−12a-\frac{1}{2}\mathbf{a}$ is half as long and points in the opposite direction.

Example

Combining scalar multiples

Given

a=(3−2),b=(−14),\mathbf{a}= \begin{pmatrix} 3\\ -2 \end{pmatrix}, \qquad \mathbf{b}= \begin{pmatrix} -1\\ 4 \end{pmatrix},

find $2a−3b2\mathbf{a}-3\mathbf{b}$ .

Multiply every component of $a\mathbf{a}$ by 2:

2a=(6−4).2\mathbf{a} = \begin{pmatrix} 6\\ -4 \end{pmatrix}.

Multiply every component of $b\mathbf{b}$ by 3:

3b=(−312).3\mathbf{b} = \begin{pmatrix} -3\\ 12 \end{pmatrix}.

Subtract corresponding components:

2a−3b=(6−4)−(−312)=(9−16).2\mathbf{a}-3\mathbf{b} = \begin{pmatrix} 6\\ -4 \end{pmatrix} - \begin{pmatrix} -3\\ 12 \end{pmatrix} = \begin{pmatrix} 9\\ -16 \end{pmatrix}.

Common Mistake

Scaling only one component

A scalar multiplies every component of the vector. For example, $3(ab)=(3a3b)3\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}3a\\3b\end{pmatrix}$ .

Scalar multiplication distributes over vector addition:

k(a+b)=ka+kb.k(\mathbf{a}+\mathbf{b})=k\mathbf{a}+k\mathbf{b}.

It also satisfies

(k+m)a=ka+ma.(k+m)\mathbf{a}=k\mathbf{a}+m\mathbf{a}.

These rules allow you to expand and simplify vector expressions in much the same way as ordinary algebra.

Parallel Vectors

Non-zero vectors are parallel when one is a scalar multiple of the other:

b=ka\mathbf{b}=k\mathbf{a}

for some scalar $k$ .

If $k$ is positive, the vectors point in the same direction. If $k$ is negative, they point in opposite directions.

Example

Checking whether vectors are parallel

Determine whether

p=(−69),q=(2−3)\mathbf{p}= \begin{pmatrix} -6\\ 9 \end{pmatrix}, \qquad \mathbf{q}= \begin{pmatrix} 2\\ -3 \end{pmatrix}

are parallel.

Compare the first components: to change 2 into negative 6, multiply by negative 3.
Check the same scalar against the second components: $−3×(−3)=9-3\times(-3)=9$ .
The same scalar works for both components, so $p=−3q\mathbf{p}=-3\mathbf{q}$ . The vectors are parallel and point in opposite directions.

Exam technique

In the exam

Keep components vertically aligned, and operate on corresponding components only.
For a vector between two points, use finish minus start and check the arrow direction.
Use the diagram as a sense check: a negative scalar reverses direction, while vector addition should represent consecutive movements.
Keep answers exact unless the question specifically asks for a decimal.

Self review

Check yourself

How would you construct $a+b\mathbf{a}+\mathbf{b}$ using the head-to-tail rule?
Given $a=(2−5)\mathbf{a}=\begin{pmatrix}2\\-5\end{pmatrix}$ and $b=(−31)\mathbf{b}=\begin{pmatrix}-3\\1\end{pmatrix}$ , can you find $3a−2b3\mathbf{a}-2\mathbf{b}$ ?
What does a negative scalar do to the magnitude and direction of a vector?

What you'll learn

Starting Point: What Is a Vector?

Adding Vectors Diagrammatically

Adding Vectors Using Components

The Zero Vector and Negative Vectors

Subtracting Vectors

Multiplying a Vector by a Scalar

Parallel Vectors

Review 1.10.4 Basic operations on vectors by teaching Genie

1.10.4 Basic operations on vectors

What you'll learn

Starting Point: What Is a Vector?

Adding Vectors Diagrammatically

Adding Vectors Using Components

The Zero Vector and Negative Vectors

Subtracting Vectors

Multiplying a Vector by a Scalar

Parallel Vectors

Review 1.10.4 Basic operations on vectors by teaching Genie

1.10.4 Basic operations on vectors

What you'll learn

Starting Point: What Is a Vector?

Adding Vectors Diagrammatically

Adding Vectors Using Components

The Zero Vector and Negative Vectors

Subtracting Vectors

Multiplying a Vector by a Scalar

Parallel Vectors

Review 1.10.4 Basic operations on vectors by teaching Genie

1.10.4 Basic operations on vectors Revision Guide

1.10.4 Basic operations on vectors

What you'll learn

Starting Point: What Is a Vector?

Adding Vectors Diagrammatically

Adding Vectors Using Components

The Zero Vector and Negative Vectors

Subtracting Vectors

Multiplying a Vector by a Scalar

Parallel Vectors

Review 1.10.4 Basic operations on vectors by teaching Genie

1.10.4 Basic operations on vectors Revision Guide