What is a vector — magnitude and direction — Mathematics

Before this

Pythagoras' theorem · coming soon
3D trigonometry
Angles of elevation and depression

This leads to

Vector addition and subtraction
Position vectors and midpoints
Displacement, velocity and acceleration as vectors · coming soon

Two friends, five kilometres, two different places

Ravi and Zainab leave the school gate at the same time. Each walks exactly 5 km. Are they now in the same place?

You cannot say. If Ravi walked north and Zainab walked east, they are about 7 km apart. If they walked the same way, they are side by side. "5 km" tells you how far. It does not tell you which way.

Some quantities need only a size. A temperature of 23 °C is complete. Other quantities are not complete until you add a direction. A wind of 30 km/h from the north and a wind of 30 km/h from the south push a boat in opposite directions. This lecture gives you a way to hold a size and a direction together, as one object you can calculate with. That object is a vector.

The 3D trigonometry lecture ended with a promise: that a direction in space would become an arrow you can calculate with. Here it is. Most of this lecture is flat, on a grid, but the last section goes back into three dimensions.

Scalars and vectors

Scalar and vector

A scalar is a quantity with size only: a number with a unit, such as 23 °C or 2 kg.

A vector is a quantity with both size and direction, such as 5 km north. The size of a vector is called its magnitude.

Physics is full of pairs that look alike but are not. Speed (60 km/h) is a scalar. Velocity (60 km/h due east) is a vector. Distance (5 km) is a scalar. Displacement (5 km north-west) is a vector. Mass is a scalar, but weight, a force, pulls in a direction, so it is a vector.

Before going on, check that you can tell them apart.

Sort — scalar or vector?

Put each quantity in the right box. Ask: does it need a direction to be complete?

This box sorts ten quantities into scalars and vectors.

Scalars (size only): a speed of 60 km/h · a distance of 5 km · a mass of 2 kg · a temperature of 23 °C · a time of 12 s
Vectors (size and direction): a velocity of 60 km/h due east · a displacement of 5 km north-west · a weight of 20 N downwards · a force of 30 N to the left · a wind of 30 km/h from the south

The pairs are the important part. A speed and a velocity can share the same number, 60 km/h, and still carry different information. The velocity holds everything the speed holds, plus a direction.

Drawing a vector, and writing one

A vector is drawn as an arrow. The length of the arrow shows the magnitude, and the arrowhead shows the direction.

There are three ways to name a vector. The arrow from point A to point B is written $A B$ , and the order of the letters matters: $B A$ points the other way. A single vector can be named with one letter in bold type, $a$ . Bold is hard to write with a pen, so when you write by hand, underline the letter instead.

The third way is the most useful for calculating. On a grid, a vector is a movement: so many squares across, and so many up or down. Write the two numbers in a column:

a = (3 - 2)

This is a column vector. The top number is the movement to the right, and the bottom number is the movement up. A negative top number means left, and a negative bottom number means down. So $a$ means "3 right, 2 down".

The three arrows labelled a start in different places, but each goes 3 right and 2 down. They are the same vector. The dashed arrow b has the same length but a different direction (3 right and 2 up), so it is a different vector.

Look at the three arrows labelled $a$ . They start in different places, but each one is the same movement: 3 right and 2 down. They are the same vector. Two vectors are equal when they have the same magnitude and the same direction, wherever they are drawn. A vector says how to move, not where to start.

That is it. That is the whole idea. A vector is a movement, and a movement can begin anywhere.

Stop and think

A student writes the vector $(3 - 2)$ as the point (3, −2). Are they the same thing?

Show the answer

No. The point (3, −2) is one fixed place on the grid. The vector is a movement of 3 right and 2 down, which can start from any point. The numbers look alike, but a point says where, and a vector says how to move. That is why vectors are written in a column, so they are not confused with coordinates.

From one point to another

Suppose A is at (2, 5) and B is at (7, 1). What is $A B$ ? To go from A to B, you move across from $x = 2$ to $x = 7$ , which is $7 - 2 = 5$ right. You move up from $y = 5$ to $y = 1$ , which is $1 - 5 = - 4$ , so 4 down.

A B = (7 - 2 1 - 5) = (5 - 4)

The rule is "finish minus start", for each coordinate. The order matters. Going back from B to A undoes the movement, so every number changes sign: $B A = (- 5 4)$ . This vector has the same length as $A B$ , but the opposite direction.

A vector that starts at the origin O has a special name. The vector $OP$ from O to a point P is called the position vector of P. If P is (4, 3), its position vector is $(43)$ . Position vectors are the one kind of vector tied to a start point, and a later lecture uses them to find midpoints.

How long is a vector?

The magnitude of a vector is the length of its arrow. Every column vector is the hypotenuse of a right-angled triangle. Its two shorter sides are the movement across and the movement up or down, as the dashed lines in the figure show. So Pythagoras gives the length.

Magnitude of a vector

The magnitude of $a = (x y)$ is written $∣ a ∣$ , and

$∣ a ∣ = x^{2} + y^{2}$

The two vertical bars are read "the magnitude of". For $a = (3 - 2)$ , $∣ a ∣ = 3^{2} + (- 2)^{2} = 9 + 4 = 13 \approx 3.61$ .

Notice the bracket round −2. The square of −2 is +4, because a length of 2 squares is 2 squares whether you move up or down. A magnitude can never be negative. It is zero only for the zero vector $(00)$ , which means "do not move".

Now explore. Change the two components and watch the arrow, its length and its direction.

Explore — build a column vector

Change the top number (across) and the bottom number (up). The arrow is drawn from two different start points to show it is the same vector. Watch the magnitude and the direction.

Top number, across: 3

Bottom number, up: −2

This table shows several column vectors, their magnitudes, and their directions, measured anticlockwise from the positive x-direction.

Vector (across, up)	Magnitude	Direction
(3, −2)	√13 ≈ 3.61	≈ 326.3°
(3, 4)	5	≈ 53.1°
(−4, 6)	2√13 ≈ 7.21	≈ 123.7°
(−5, 0)	5	180°

Changing the sign of a component flips the arrow but keeps its length, because the square of a negative number is positive.

You should have seen three things. Changing a sign flips the arrow but never changes its length. Swapping the two numbers gives a different direction with the same length. And the two arrows always match, because a vector does not depend on where it starts.

Worked examples

Worked Example 1 — Reading a column vector

Describe $v = (- 4 6)$ , find its magnitude, and find its direction as an angle measured anticlockwise from the positive x-direction.

Think first. Read the signs: −4 across means 4 left, and 6 up means 6 up. So the arrow points up and to the left.

$∣ v ∣ = (- 4)^{2} + 6^{2} = 16 + 36 = 52$ . Since $52 = 4 \times 13$ , $52 = 213 \approx 7.21$ .

For the direction, sketch the right-angled triangle: 4 across and 6 up. The angle it makes with the negative x-direction is $tan^{- 1} (6/4) \approx 56. 3^{\circ}$ . The arrow is that far above the leftward direction, so the angle from the positive x-direction is $18 0^{\circ} - 56. 3^{\circ} = 123. 7^{\circ}$ .

Check. The arrow points up and left, so its angle must be between 90° and 180°. ✓ And $∣ v ∣$ is longer than both 4 and 6, as a hypotenuse must be. ✓

Worked Example 2 — The vector between two points

A is (2, 5) and B is (7, 1). Find $A B$ and the distance AB.

Think first. Finish minus start, for each coordinate. Then the magnitude is the distance.

$A B = (5 - 4)$ , as shown earlier. So $∣ A B ∣ = 25 + 16 = 41 \approx 6.40$ .

Check. The distance formula from coordinate geometry, $(7 - 2)^{2} + (1 - 5)^{2}$ , is exactly the same calculation. ✓ The distance formula was always a vector magnitude in disguise.

Worked Example 3 — Finding a missing component

The vector $(k 12)$ has magnitude 13. Find the possible values of $k$ .

Think first. Write the magnitude as an equation, then solve it.

$k^{2} + 144 = 13$ , so $k^{2} + 144 = 169$ and $k^{2} = 25$ . So $k = 5$ or $k = - 5$ .

Check. Both $(512)$ and $(- 5 12)$ have length $169 = 13$ . ✓ There are two answers because the arrow can lean right or left and keep the same length.

Back into three dimensions

The 3D trigonometry lecture found the longest line in a box with $d^{2} = a^{2} + b^{2} + c^{2}$ . Vectors carry this over directly. In space, a movement has three parts: across, along and up. So a vector in three dimensions has three numbers in its column:

w = 2 - 3 6

Its magnitude uses all three: $∣ w ∣ = 2^{2} + (- 3)^{2} + 6^{2} = 4 + 9 + 36 = 49 = 7$ .

This is the space diagonal of a box 2 by 3 by 6, written as one arrow. The same idea works for any number of directions, which is why vectors are used for everything from forces to computer graphics.

Three traps

Trap 1 — Start minus finish

$A B$ is B's coordinates minus A's: finish minus start. Using A minus B gives $B A$ , the same length but the opposite direction. Check the signs with a sketch: if B is to the right of A, the top number must be positive.

Trap 2 — Adding the components for the length

The vector $(3 - 2)$ does not have length $3 + 2 = 5$ , or $3 - 2 = 1$ . The length is the hypotenuse, $3^{2} + (- 2)^{2} = 13$ . Always square first, with the sign inside a bracket: $(- 2)^{2} = 4$ , but $- 2^{2}$ on a calculator gives −4.

Trap 3 — A vector is not a point

The column vector $(3 - 2)$ is a movement that can start anywhere. The point (3, −2) is one fixed place. Only a position vector, which starts at O, matches the coordinates of a point.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Write each movement as a column vector and find its magnitude, exactly where you can: (a) 6 left and 8 up; (b) 5 right and 3 down; (c) 7 down.

Hint 1
Left and down give negative numbers. "7 down" has no movement across.

Hint 2
Use $x^{2} + y^{2}$ , with each negative number in a bracket.

Full answer
(a) $(- 6 8)$ , magnitude $36 + 64 = 100 = 10$ . (b) $(5 - 3)$ , magnitude $25 + 9 = 34 \approx 5.83$ . (c) $(0 - 7)$ , magnitude 7. Check: (c) moves straight down 7, so its length must be 7. ✓
P is (−3, 4) and Q is (5, −2). Find $PQ$ , $QP$ and the distance PQ.

Hint 1
Finish minus start: for $PQ$ , Q's coordinates minus P's.

Hint 2
$5 - (- 3) = 8$ .

Full answer
$PQ = (5 - (- 3) - 2 - 4) = (8 - 6)$ . $QP = (- 8 6)$ . Distance $= 64 + 36 = 100 = 10$ . Check: Q is to the right of P and below it, so $PQ$ has a positive top number and a negative bottom number. ✓
The vector $(k 2 k)$ has magnitude $45$ . Find the possible values of $k$ , and write down each vector.

Hint 1
$k^{2} + (2 k)^{2} = 45$ .

Hint 2
$(2 k)^{2} = 4 k^{2}$ , not $2 k^{2}$ .

Full answer
$k^{2} + 4 k^{2} = 5 k^{2} = 45$ , so $k^{2} = 9$ and $k = 3$ or $k = - 3$ . The vectors are $(36)$ and $(- 3 - 6)$ . Check: $9 + 36 = 45$ . ✓ The two vectors point in exactly opposite directions.
A stretch. A drone takes off from a point O on flat ground. It flies in a straight line to a point 3 km east, 4 km north and 1.2 km up. Take east, north and up as the three directions. (a) Write its displacement as a three-number column vector, in km. (b) Find how far it is from O. (c) Find the angle its path makes with the ground.

Hint 1
(b) Use $x^{2} + y^{2} + z^{2}$ .

Hint 2
(c) As in the 3D trigonometry lecture, the angle with the ground is measured against the shadow of the path on the ground, which has length $3^{2} + 4^{2}$ .

Full answer
(a) $34 1.2$ . (b) $9 + 16 + 1.44 = 26.44 \approx 5.14$ km. (c) The shadow on the ground has length $9 + 16 = 5$ km, so $tan θ = 1.2 \div 5 = 0.24$ , and $θ \approx 13. 5^{\circ}$ . Check: $sin θ = 1.2 \div 5.142 \approx 0.2334$ also gives about 13.5°. ✓

The bigger idea

The idea of a vector took decades to settle. In 1835 the Italian mathematician Giusto Bellavitis treated equal, parallel arrows as one object, which is the idea of the three arrows labelled $a$ in this lecture. The word "vector" was coined by William Rowan Hamilton around 1843. The system used today was built in the 1880s by Josiah Willard Gibbs and, independently, Oliver Heaviside. In 1891 Heaviside argued for printing vectors in bold type, as this lecture does (sources: Wikipedia, "Vector notation"; M. J. Crowe, A History of Vector Analysis).

It mattered because physics needed it. Forces, velocities and fields all have directions, and they combine in ways that plain numbers cannot describe. Walk 3 km east and then 4 km north, and you are 5 km from the start, not 7. Adding vectors is the next lecture.

Key concepts in this lesson

A scalar has size only; a vector has size (its magnitude) and direction.
A vector can be written $A B$ , as a bold letter such as $a$ (underlined by hand), or as a column vector with the movement across on top and the movement up below.
Two vectors are equal when they have the same magnitude and direction, wherever they are drawn.
The vector from A to B is found by "finish minus start" for each coordinate, and $B A$ has every sign reversed.
The magnitude of $(x y)$ is $x^{2} + y^{2}$ by Pythagoras, and in three dimensions $x^{2} + y^{2} + z^{2}$ .
A position vector starts at the origin O and gives the position of a point.