9.1 Introduction

Many things can be represented by a simple number, for instance, time, distance, mass, which are then called scalar quantities. Others, however, are better represented by both their size, or magnitude, and a direction. Some of these are velocity, acceleration, and force. These quantities are called vector quantities because they are represented by vectors.

A simple example of a vector is one that describes displacement. Supposing someone is standing in a room with floor tiles (as in Figure 9.1) and moving from one position to another can be described by the number of tiles to the right and the number of tiles towards the top of the page.

In the example, to move from the door to the cupboard can be represented by (4, 2). This vector consists of two numbers, where the order of the numbers is important. Moving (4, 2) results in a different final position to that if we move (2, 4). The magnitude of the displacement can be found by drawing a straight line from the starting position to the final position and measuring the length. From Pythagoras theorem this can be found as $\sqrt{4^2?+?2^2}?=?\sqrt{20?\approx ?4.47}$. The direction can be described by an angle, for instance, the angle made to the wall with the window on it.

This example shows that a two-dimensional vector (2D) can be used to represent movement on a flat surface. A 2D vector is two numbers, where the order of the numbers is important.

If the room in Figure 9.1 also had wall tiles then we could represent a position above the floor by the number of tiles towards the ceiling. This three-dimensional (3D) vector can be represented by three numbers. It can also be represented by the distance travelled and the direction, angles made to the floor and the angle made to the wall.

Velocity is an example of a vector quantity. This can be described by two things, the speed, which is the rate of change of distance travelled with respect to time, and also the direction in which it is travelling. Similarly, force can be described by the size of magnitude of the force and also the direction in which it operates.

Vectors have their own rules for addition and subtraction. If two forces of equal magnitude operate on one object then the net effect will depend on the direction of the forces. If the forces operate in opposite directions they could balance each other out, like two tug-of-war teams in a stalemate struggle. Alternatively, they could operate in the same direction or partially in the same direction and cause the object to have an acceleration.

For the examples of vectors given so far, the maximum dimension of the vector is three as there are only three spatial dimensions. However, there are many examples when vectors of higher dimension are useful. For instance, a path through the network given in Figure 9.2 can be represented by a list of 1 s and 0s to indicate whether each of the edges is included in the path. A path from S to T can be represented by a vector, for instance:

$\begin{array}{lllllllll}a\hfill & b\hfill & c\hfill & d\hfill & e\hfill & f\hfill & g\hfill & h\hfill & \hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & \text{represents}?\text{the}?\text{path}?be\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & \text{represents}?\text{the}?\text{path}?cfg.\hfill \end{array}$

Although there are many other types of vectors we will concentrate on vectors of two or three dimensions, called spatial or geometrical vectors, used to represent physical quantities in space. Many of the ideas in this chapter are only true for geometrical vectors of two and three dimensional. As 3D vectors can only be correctly represented by making a 3D model, it is important to concentrate on understanding 2D vectors as they can be drawn on a piece of paper allowing results to be checked easily.

9.2 Vectors and vector quantities

A vector is a string of numbers, for example,

(1,2, −1)

(1,0)

(3, −4, 2, −6, 8)

(2.6,9, −1.2,0.3).

The length of the string is called the dimension of the vector. For the examples given above, the dimensions are 3, 2, 5, and 4, respectively. The commas can be left out, so the examples given above can be written as

(1 2 −1)

(1 0)

(3 −42 −6 8)

(2.69 −1.20.3).

Vectors may also be written as columns, giving:

$\left(\begin{array}{c}1\\ 2\\ -1\end{array}\right)?\left(\begin{array}{c}1\\ 0\end{array}\right)?\left(\begin{array}{c}3\\ -4\\ 2\\ -6\\ 8\end{array}\right)?\left(\begin{array}{c}2.6\\ 9\\ -1.2\\ 0.3\end{array}\right).$

Whether vectors are written as columns or rows only becomes important when we look at matrices (Chapter 13). However, the order of the numbers in the vector is important: (0, 1) is a different vector from (1, 0).

We will mainly deal with 2D or 3D vectors. Vectors are represented in a diagram by a line segment with an arrow as in Figure 9.3. In printed material vectors can be represented by bold letters: a. They are also represented by a or $\overrightarrow{a}$ or $\overrightarrow{AB}$, where A and B are points at either end of the vector.

In the rest of this chapter, we will assume that we are dealing with 2D or 3D vectors represented in rectangular form, also called Cartesian form.

This means that the numbers in the vectors correspond to the x, y, z values for a set of rectangular axes, as shown in Figure 9.4. This assumption is important for many of the geometrical interpretations presented here.

Position vectors and translation vectors

Vectors can represent points in a plane, as in Figure 9.5(a), or points in space, as in Figure 9.5(b). These are called position vectors. They can be thought of as representing a translation from the origin.

Vectors can represent a translation that can be applied to figures. In Figure 9.6, a four-sided figure ABCD has been translated through the vector (2, 3).

Vector quantities

Vectors can represent physical quantities that have both a magnitude and a direction. In Figure 9.7, there is an example of the forces acting on a body that is being pulled up a slope. By using vectors and vector addition the resultant force acting on the body can be found and therefore the direction in which the body will travel can be found together with the size of the acceleration. Other quantities with both magnitude and direction are velocity, acceleration, and moment. Quantities that only have magnitude and no direction are called scalar quantities and can be represented using a number, for example, mass and length.

9.4 Magnitude and direction of a 2D vector–polar co-ordinates

We have already noted that a vector has magnitude and direction. A 2D vector can be represented by its length (also called magnitude or modulus), r (or |r|), and its angle to the x-axis, also called its argument, θ. If the vector is (x, y) then r²= x² + y², from Pythagoras's theorem The angle is given by tan⁻¹ (y / x) if x is positive and by tan⁻¹ (y / x) + π if x is negative. Hence, r = (r, θ) in polar coordinates, and can also be written as r ∠θ so it is clear that the second number represents the angle. As it is usual to give the angle between –π and +π, it may be necessary to subtract 2π from the angle given by this formula. (As a rotation of 2π is a complete rotation this will make no difference to the position of the vector.)

Example 9.4

Find the magnitude and direction of

(a) (2, 3)

(b) (–1, – 4)

(c) (1, −2.2)

(d) (–2, 5.6)

Solution To perform these conversions to polar form it is a good idea to draw a diagram of the vector in order to be able to check that the angle is of the correct size. Figure 9.11 shows the diagrams for each part of the example.

(a) r = (2, 3) has magnitude $\sqrt{2^2+3^2}$ ≈ 3.606 and angle given by tan⁻¹(3/2) ≈ 0.983 and therefore in polar coordinates r is 3.606 ∠0.983.

(b) r = (−1, −4) has magnitude r = $r=\sqrt{{\left(-1\right)}^2+{\left(-4\right)}^2}\approx 4.123$, and angle given by tan⁻¹(−4/−l)+π ≈1.326+3.142 = 4.467. As this angle is bigger than π, subtract 2π (a complete revolution) to give −1.816. Therefore, in polar co-ordinates r = 4.123 ∠ −1.816. Note that the angle is between –π and − π /2, meaning that the vector must lie in the third quadrant, which we can see is correct from the diagram.

(c) r = (1, −2.2) has magnitude $\text{r}=\sqrt{{\left(1\right)}^2+{\left(-2.2\right)}^2}\approx 2.416$ and the angle is given by tan⁻¹ (−2.2/1) ≈ −1.144. Therefore, in polar co-ordinates r = 2.416 ∠−1.144. Note that the angle is between −π /2 and 0, meaning that the vector must lie in the fourth quadrant.

(d) r = (−2,5.6) has magnitude $r=\sqrt{{\left(-2\right)}^2+{\left(5.6\right)}^2}\approx 5.946$ and angle given by tan^−l (5.6/ – 2) +π ≈1.914. Therefore, in polar co-ordinates r = 5.946 ∠1.914. Note that the angle is between π/2 and π, meaning that the vector must lie in the second quadrant.

Many calculators have a rectangular to polar conversion facility. Look this up on the instructions with your calculator and check the results. Remember, to get the result in radians you should first put your calculator into radian mode.

Conversion from polar co-ordinates to rectangular co-ordinates

If a vector is given by its length and angle to the x-axis, that is, r = r ∠θ, then

$\begin{array}{cc}x=r?cos\left(\theta \right)& y=r?sin\left(\theta \right)\end{array}$

Hence, in rectangular co-ordinates r = (r cos (θ), r sin (θ)).

This result can easily be found from the triangle, as shown in Figure 9.12; examples are given in Figure 9.13.

9.5 Application of vectors to represent waves (phasors)

In Section 5.3, we found the amplitude, phase, and cycle rate (frequency) of a wave. f(t) = A cos(ωt + ϕ) has amplitude A, angular frequency ω and phase ϕ. Suppose we consider waves of a fixed frequency (say 50 Hz giving ω = 50 × 2π ≈ 314); then, different waves can be represented by the amplitude and phase, giving y = A∠ϕ. The ideas of vectors can then be used to add and subtract waves and find their combined effect.

If a wave can be represented in polar form by A∠ϕ, then what does the rectangular form of the vector represent? We find that if the wave is split into cosine and sine terms by using the trigonometric identity cos(A + B) = cos(A) cos(B) – sin(A) sin(B), we get:

f(t) = A cos (ωt + ϕ) = A cos (ϕ) cos (ωt) — A sin (ϕ) sin (ωt).

As A cos (ϕ) is a constant, not involving an expression in t, this can be replaced by c and similarly A sin (ϕ) can be replaced by d giving

f(t) = c cos(ωt) — d sin(ωt)

where c = A cos(ϕ) and d = A sin(ϕ).

So the vector (c, d) used to represent a wave represents the function f(t) = c cos(ωt) — d sin(ωt) and if expressed in polar form A∠ϕ it represents the equivalent expression

f(t) = A cos(ωt+ϕ)

9.6 Multiplication of a vector by a scalar and unit vectors

Multiplying a vector by a scalar has the effect of changing the length without affecting the direction. Each number in the vector is multiplied by the scalar

Example 9.11

If a = (1, −2) then

3a = 3 (1, −2) = (3 × 1,3 × (–2)) = (3, −6)

0.5a = 0.5(1, −2) = (0.5 × 1,0.5 × (−2)) = (0.5, −1).

This is shown in Figure 9.14.

9.7 Basis vectors

Vectors in a plane are made up of a part in the x-direction and a part in the y-direction, for example, (2,3) = (2,0) + (0,3). i and j are used to represent unit vectors in the x-direction and y-direction, that is, i = (1,0) and j = (0, 1).

Any vector in the plane can be expressed in terms of i and j. i and j are called the Cartesian unit basis vectors, which is the name given to a co-ordinate system where the axes are at right angles to each other (orthogonal) (see Figure 9.16):

(2,3) = 2(1,0) + 3(0, 1) = 2i + 3j.

The unit vector in the z direction is often given the symbol k; and in three dimensions, using rectangular axes we have:

i = (1, 0, 0) j = (0, 1, 0) k = (0, 0, 1)

that is

(5, −1,2) = (5,0,0) + (0, −1,0) + (0,0,2)

= 5(1,0,0) + (–1)(0,1,0) + 2(0,0,1)

= 5i – j + 2k.

The vectors i and j form a basis set because all 2D geometrical vectors can be expressed in such terms. Similarly, all 3D vectors can be expressed in terms of i, j, and k. There are many other sets of vectors that can be used as a basis set: for instance, if we were in a room shaped like a parallelogram we could express any position in the room by moving parallel to one of the sides and then parallel to the other side. This is shown in Figure 9.17. Other basis sets are not as useful for interpreting spatial vectors as they do not give the same geometrical results. For instance, the interpretation of the scalar product, given in Section 9.8, relies on the fact that we use Cartesian basis vectors.

9.8 Products of vectors

There are two products of vectors that are commonly used: the scalar product, which results in a scalar, and the vector or cross product, which gives a vector as the result. However, both of these products are irreversible: they have no inverse operation. In other words, it is not possible to divide by a vector.

9.9 Vector equation of a line

In Chapter 2, we looked at the equation of a line that we found to be y = mx + c, where the gradient of the line is m and the line goes through the point (0, c). We also found that the equation of a line that goes through two points, (X_l, Y₁) and (x₂, y₂), is

$\frac{y?-?y_1}{y_2?-?y_1}?=?\frac{x?-?x_1}{x_2?-?x_1}.$

We would like to be able to express the equation of a line as a vector equation. If we know that the two points A and B represented by the position vectors a and b lie on the line, then a vector in the direction of the line will be a vector joining those two points, that is, b – a. As the line must go through A, we can see that any multiple of b – a added to the position vector a must lie on the line. This is shown in Figure 9.23

If we call the position vector of any point on the line r where r = (x, y), we now have the vector equation of the line as r = a + λ (b – a) where λ ∈ .

This can be rewritten as r = a (1 – λ) + λb.

9.10 Summary

1. A vector is a string of numbers where the length of the string is called the dimension of the vector.

2. Vectors are used to represent points on a plane or in space, translations and physical quantities that have both magnitude and direction (called vector quantities).

3. The vector sum is found by adding corresponding elements of the vectors, or from a diagram by using a parallelogram.

4. To subtract vectors, subtract corresponding elements of the vectors. A triangle may be used to perform vector subtraction in a diagram.

5. Two-dimensional vectors r = (x, y) can be expressed in polar co-ordinates using

$r?=?\sqrt{x^2?+?y^2}$

and θ = tan⁻¹(y) (+π, if x is negative), so that (x, y) = r∠θ where r or |r|is the magnitude, or length, of the vector and θ is theangle that the vector makes to the x-axis, also called its argument. To convert from polar to rectangular co-ordinates use:

$x?=?r?cos\left(\theta \right)?\text{and}?y?=?r?sin\left(\theta \right).$

To add vectors given in polar form they must first be converted to rectangular form.

6. Waves of a fixed frequency can be represented by phasors giving the amplitude and phase. f(t)= A cos (ωt + ϕ) can be represented by its amplitude and phase A∠ϕ. Converting this vector to rectangular form gives (c,d) where

$f\left(t\right)?=?c?cos\left(\omega t\right)?-?d?sin\left(\omega t\right).$

Using ideas of conversion from polar to rectangular form and vector addition, waves of the same frequency can be easily combined.

7. Unit vectors have length 1. To find the unit vector in the same direction as a vector r divide the vector by its length:

$\hat{\text{r}}?=?\frac{\text{r}}{\left|\text{r}\right|}.$

8. Any vectors in the plane can be represented in terms of i = (1, 0) and j = (0,1) and in three dimensions by i = (1,0, 0), j = (0,1,0), and k = (0,0, 1). These are the Cartesian unit basis vectors and they are at right angles to each other.

9. Where a = (a₁, a₂) and b = (b₁,b₂)are two vectors, the scalar product is given by a · b = (a₁a₂) ·(b_l, b₂) = a₁b₁+ a₂b₂and a ·b = ab cos(θ), where a, b are the magnitudes of the vectors a and b, and θ is the angle between them. The scalar product can be used to find the angle between two vectors.

10. The components of any unit vector give the cosines of the angles that the vector makes to each of the Cartesian axes. Then we have for any vector r:

$\hat{\text{r}}?=?\frac{\text{r}}{\left|\text{r}\right|}?=?\left(cos\left(\alpha \right),?cos\left(\beta \right)\right)$

where α is the angle the vector r makes to the x-axis and β is the angle it makes to the y-axis. In three dimensions:

$\hat{\text{r}}?=?\frac{\text{r}}{\left|\text{r}\right|}?=?\left(cos\left(\alpha \right),?cos\left(\beta \right),?cos\left(\gamma \right)\right)$

where α is the angle the vector r makes to the x-axis, β is the angle it makes to the y-axis, and γ is the angle it makes to the z-axis. cos (α), cos (β), and cos (γ) are called the direction cosines of the vector.

11. The scalar product can be used to find the component of a vector in any given direction. Component of a vector F in the direction of a vector r = F · (r /|r|).

12. The vector product is given by a × b = ab sin (θ)$\hat{\text{n}}$, where a and b are the magnitudes of the vectors a and b, θ is the angle between them, and $\hat{\text{n}}$ is a unit vector normal to the plane of a and b. If a and b are vectors in the x, y plane, that is, a = (a₁, a₂, 0) and b = (b_l, b₂, 0), we have
(a₁, a₂, 0) × (b_l, b₂, 0) = (0,0, a₁b₂ –a₂b₁) = (a_lb₂ – a₂b₁)k
where k is the unit vector in the z-direction. The magnitude of the vector product can be used to find the area of a parallelogram.

13. The vector equation of a line passing through two points a and b is

r = a(1 – λ) + λb λ ∈

9.12 Exercises

9.1. In Figure 9.24, $\overrightarrow{OA}$ = a and $\overrightarrow{OB}$ = b, OA = BC = OD, OB = AC = EO, EOB and DOA are straight lines. Write the following in terms of a and b:

(a) $\overrightarrow{AB}$

(b) $\overrightarrow{BA}$

(c) $\overrightarrow{OC}$

(d) $\overrightarrow{OE}$

(e) $\overrightarrow{OD}$

(f) $\overrightarrow{ED}$

(g) $\overrightarrow{DE}$

(h) $\overrightarrow{DA}$

(i) $\overrightarrow{BE}$

(j) $\overrightarrow{EA}$

9.2. Given a = (1,3), b = (–1,2), C = (3,6,2), and d = (6,4, −1), find the following:

(a) a + b

(b) a – b

(c) b – a

(d) − b + a

(e) 2b

(f) a + 2b

(g) 3a – b

(h) c –d

(i) 10c

(j) c + 6d

(k) 6c – d

9.3. Express the following in its polar form, r ∠θ, where r is the length of the vector and θ its angle to the x-axis:

(a) (1,3)

(b) (3, −1)

(c) (−1, −3)

(d) (5, −6)

9.4. Express the following vectors r ∠θ, where r is the modulus of the vector and θ the angle to the x-axis, in rectangular form. The angle is expressed in radians.

(a) 5∠π

(b) l ∠ –π

(c) $\frac{1}{2}$∠π/4

(d) 3∠π/3

9.5. Express the following as a sum of a cosine and sine term in ωt:

(a) f(t) = 3 cos(3t – 2)

(b) f(t) = 10 cos(20t + 5)

9.6. Express the following as single cosine terms:

(a) f(t) = 4 cos(10t) – 3 sin(10t)

(b) g(t) = −2 cos(157t)+ 10 sin(157t)

9.7. Express the following as a single wave:

(a) 6 cos(2t –3) + 10 cos(2t+ 2)

(b) cos(t –π /2) + cos(t + π /2)

(c) 2 cos(628t – 1.57) −6 cos(628t)

9.8. Find the unit vectors in the same direction as the following:

(a) (6, 8)

(b) (5, 12)

(c) (5, – 12)

(d) (1, 1)

(e) (3,2)

(f) (2,0)

(g) (0, −3)

(h) (2,4,4)

(i) (1, −1,2)

(j) (0.5,0, −0.5)

9.9. Express the following vectors in terms of i = (1,0) and j = (0, 1) or in terms of i = (0, 0, 1), j = (0, 1, 0), and k = (0, 0, 1) for 3D vectors:

(a) (5,2)

(b) (–1, −2)

(c) (–6,2)

(d) (–1,2, −3)

(e) (0.2, −1.6,3.3)

9.10. Find the following scalar products:

(a) (1, – 2) · (3, 3)

(b) (9, 2) · (– 1, 6)

(c) (6, −1) · (−1, −3)

9.11. Find the angle between the following pairs of vectors:

(a) (1, −2) and (5, 1)

(b) (6, −1) and (1,6)

(c) (2, −1) and (4,9)

9.12. Show that the following pairs of vectors are at right angles to each other:

(a) (2, 1) and (−1, 2)

(b) (–6,3) and (1,2)

(c) (0.5, −2) and (4, 1)

9.13. Find the angles that the following vectors make to the axes:

(a) (3,6)

(b) (−1, −4,5)

9.14. (a) Find the component of the vector (−1,5) in the direction of the following vectors:

(i) (0.5,0.5)

(ii) (0.5, −0.5)

(iii) (−5, 1)

(iv) (1, −5)

(v) (8,2)

(b) Find the component of the vector (−1, 2, 7) in the direction of the following vectors:

(i) (1, 1, 1)

(ii) (6, 0, 2)

9.15. Show that the following pairs of vectors are parallel:

(a) (−3,1) and (1.5, −0.5)

(b) (6,3) and (18,9)

9.16. Find the area of the parallelogram OABC where two adjacent sides are:

(a) $\overrightarrow{OA}$= (1,−1) and $\overrightarrow{OC}$= (5,2)

(b) $\overrightarrow{OA}$= (4, −1) and $\overrightarrow{OC}$= (2,2)

(c) $\overrightarrow{OA}$= (−3, 1) and $\overrightarrow{OC}$ = (2,3)

9.17. A straight line passes through the pair of points given. Find the vector equation of the line in each case:

(a) (0,1), (−1,4)

(b) (1,1), (−2, −4)

(c) (1,1), (6,3)

(d) (−1, −4), (−3, −4)