A vector is a quantity that has a magnitude and a direction (as opposed to a scalar that only has a magnitude). Vectors are used in many areas of science and mathematics. This module introduces vectors and describes how to add, subtract and multiply them.

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Vectors and scalars

One example of a vector is velocity . The velocity of an object is determined by the magnitude (speed) and direction of travel. Other examples of vectors are force, displacement and acceleration.

A scalar is a quantity that has magnitude only. Mass, time and volume are all examples of scalar quantities.

Vectors in three-dimensional space are defined by three mutually perpendicular directions and and can be denoted in a number of ways, most usually with an arrow above  [latex]\vec{a}[/latex]  or  [latex]\vec{b}[/latex]  or  [latex]\vec{c}[/latex]  or  with a tilde below  [latex]\underset{\sim}{u}[/latex]  or   [latex]\underset{\sim}{v}[/latex].

A vector in the opposite direction from  [latex]\vec{a}[/latex]  is denoted by  [latex]-\vec{a}[/latex].

Vectors can be added or subtracted graphically using the triangle rule.

Adding and subtracting vectors

Triangle Rule:

To add vectors [latex]\vec{a}[/latex] and [latex]\vec{b}[/latex] shown above, place the tail of vector [latex]\vec{b}[/latex] at the head of vector [latex]\vec{a}[/latex] (shown below).

The vector sum, [latex]\vec{a}+\vec{b}[/latex], is the vector [latex]\overrightarrow{AC}[/latex] , from the tail of vector [latex]\vec{a}[/latex] to the head of [latex]\vec{b}[/latex].

To subtract [latex]\vec{b}[/latex] from [latex]\vec{a}[/latex], reverse the direction of [latex]\vec{b}[/latex] to give [latex]-\vec{b}[/latex] then add [latex]\vec{a}[/latex] and [latex]-\vec{b}[/latex].

[latex]\vec{a}-\vec{b}=\vec{a}+(-\vec{b})[/latex]

Vector [latex]\overrightarrow{AC}[/latex] is equal to the vector [latex]\vec{a}-\vec{b}[/latex].

Components of a vector

The diagram below shows the vector [latex]\overrightarrow{OP}[/latex], where [latex]O[/latex] is the origin and [latex]P[/latex] is the point [latex](1,2,3)[/latex].

[latex]\vec{i}[/latex] , [latex]\vec{j}[/latex] , and [latex]\vec{k}[/latex] are vectors of magnitude one in the direction of the [latex]x[/latex] , [latex]y[/latex], and [latex]z[/latex] axes, respectively.

A vector of magnitude one is called a unit vector.

[latex]1\vec{i}[/latex] is a vector of length [latex]1[/latex] in the direction of the [latex]x[/latex]-axis.

[latex]2\vec{j}[/latex] is a vector of length [latex]2[/latex] in the direction of the [latex]y[/latex]-axis.

[latex]3\vec{k}[/latex] is a vector of length [latex]3[/latex] in the direction of the [latex]z[/latex]-axis.

[latex]\overrightarrow{OP}[/latex] is then the vector [latex]\vec{i}+2\vec{j}+3\vec{k}[/latex].

[latex]1\vec{i}[/latex], [latex]2\vec{j}[/latex] and [latex]3\vec{k}[/latex] are called the components of the vector.

The vector [latex]\overrightarrow{OP}[/latex] is denoted by [latex]\vec{i}+2\vec{j}+3\vec{k}[/latex] but may also be written as [latex]\left(1,2,3\right)[/latex].

In general, the notation [latex](x,y,z)[/latex] is often used to denote the vector [latex]\left(x\vec{i}+y\vec{j}+z\vec{k}\right)[/latex] as well as the co-ordinates of a point [latex]P (x,y,z)[/latex]. The context will determine the correct meaning, as [latex](x,y,z)[/latex] might represent either a point or a vector.

Vectors can be added geometrically (head to tail) but they may also be added or subtracted arithmetically by adding or subtracting their corresponding components.

Now try the following exercise.

Directed line segment

The directed line segment, or geometric vector, [latex]\overrightarrow{PQ}[/latex], from a point  [latex]P(x_{1},y_{1},z_{1})[/latex] to a point [latex]Q(x_{2},y_{2},z_{2})[/latex] is found by subtracting the co-ordinates of [latex]P[/latex] (the initial point) from the co-ordinates of [latex]Q[/latex] (the final point).

[latex]\overrightarrow{PQ}=(x_{2}-x_{1})\vec{i}+(y_{2}-y_{1})\vec{j}+(z_{2}-z_{1})\vec{k}[/latex]

Example: Directed line segments

The directed line segment [latex]\overrightarrow{PQ}[/latex] in Fig. 17.4 is represented by the vector [latex]\vec{i}+\vec{j}-2\vec{k}[/latex], or [latex](1,1,-2)[/latex]. Any other directed line segment with the same length and same direction as [latex]\overrightarrow{PQ}[/latex] is also represented by [latex]\vec{i}+\vec{j}-2\vec{k}[/latex] or [latex](1,1,-2)[/latex].

The directed line segment [latex]\overrightarrow{QP}[/latex] has the same length as [latex]\overrightarrow{PQ}[/latex] but is in the opposite direction.

[latex]\overrightarrow{QP} =-\overrightarrow{PQ} =-(\vec{i}+\vec{j}-2\vec{k}) =-\vec{i}-\vec{j}+2\vec{k} =(-1,-1,2).[/latex]

Position vectors

The position vector of any point is the directed line segment from the origin [latex]O (0,0,0)[/latex] to that point and is given by the co-ordinates of the point.

For example, the position vector of [latex]P(3,4,1)[/latex] is [latex]3\vec{i}+4\vec{j}+\vec{k}[/latex], or [latex](3,4,1)[/latex].

Magnitude of a vector

If  [latex]\vec{a}=a_{1}\vec{i}+a_{2}\vec{j}+a_{3}\vec{k}[/latex]  the length or magnitude of [latex]\vec{a}[/latex] is written as [latex]\left|\vec{a}\right|[/latex] and is evaluated as:

[latex]\left|\vec{a}\right|=\sqrt{(a_{1})^{2}+(a_{2})^{2}+(a_{3})^{2}}.[/latex]

Unit vector

Any vector with a magnitude of one is called a unit vector.

If [latex]\vec{a}[/latex] is any vector, then a unit vector parallel to [latex]\vec{a}[/latex] is written [latex]\hat{a}[/latex] (we call this [latex]a[/latex] “hat”). The “hat” symbolises a unit vector.

The unit vector [latex]\hat{a}[/latex] equals the vector [latex]\vec{a}[/latex] divided by its magnitude [latex]\left|\vec{a}\right|[/latex] . That is

\begin{align*}
\hat{a} &=\dfrac{\vec{a}}{\left|\vec{a}\right|}.
\end{align*}

Rearanging this formula gives

\begin{align*}\vec{a} =\left|\vec{a}\right|\hat{a}. \end{align*}

Of particular importance are the unit vectors [latex]\vec{i}[/latex], [latex]\vec{j}[/latex] and [latex]\vec{k}[/latex] (sometimes written as [latex]\hat{i}[/latex], [latex]\hat{j}[/latex] and [latex]\hat{k}[/latex]), parallel to the [latex]x[/latex], [latex]y[/latex], and [latex]z[/latex] axes, respectively.

Now attempt the questions in the exercise below.

Multiplication by a scalar

To multiply a vector [latex]\vec{a}=a_{1}\vec{i}+a_{2}\vec{j}+a_{3}\vec{k}[/latex] by a scalar, [latex]m[/latex], multiply each component of [latex]\vec{a}[/latex] by [latex]m[/latex]:

\begin{align*}m\,\vec{a} =m\,a_{1}\vec{i}+m\,a_{2}\vec{j}+m\,a_{3}\vec{k}.\end{align*}

The result is a vector of length [latex]m\times\left|\vec{a}\right|=m\left|\vec{a}\right|.[/latex]

If [latex]m>0[/latex], the resultant vector is in the same direction as [latex]\vec{a}.[/latex]

If [latex]m\lt{0}[/latex], the resultant vector is in the opposite direction to [latex]\vec{a}.[/latex].

Two vectors [latex]\vec{a}[/latex] and [latex]\vec{b}[/latex] are said to be parallel if and only if  [latex]\vec{a}=c\vec{b}[/latex] where [latex]c[/latex] is a real constant.

Try the questions below.

Multiplication of vectors

1. The scalar product (also known as the dot product) which gives a number.
2. The vector product (also known as the cross product) which gives a vector.

Scalar product

[latex]\vec{a}×\vec{b}=[/latex] [latex]\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{array}\right|[/latex]

[latex]=\left|\begin{array}{cc} a_{2} & a_{3}\\ b_{2} & b_{3} \end{array}\right|[/latex][latex]\hat{i}[/latex] [latex]-\left|\begin{array}{cc} a_{1} & a_{3}\\ b_{1} & b_{3} \end{array}\right|[/latex][latex]\hat{j}[/latex] [latex]+\left|\begin{array}{cc} a_{1} & a_{2}\\ b_{1} & b_{2} \end{array}\right|[/latex][latex]\hat{k}[/latex]