Scalars and vectors

Seven kilometres, or five?

You leave the school gate and walk $3\ \text{km}$ east. Then you turn and walk $4\ \text{km}$ north. Your step counter says you walked $7\ \text{km}$. A map says you are $5\ \text{km}$ from the gate. Which one is right?

Both are right. They answer two different questions.

• How far did I walk? That is a plain number: $7\ \text{km}$.
• Where am I now, compared with the gate? That needs a number and a direction: $5\ \text{km}$, towards the north-east.

The first kind of quantity is a scalar. The second kind is a vector. This whole chapter is built on that difference, so let us look at it carefully.

Scalars: a size and nothing more

A scalar has only a magnitude, which means a size. You give a number and a unit, and you are done.

Distance, speed, mass, time, temperature, energy and volume are scalars. Two scalars of the same kind add like ordinary numbers. A $2\ \text{kg}$ bag and a $3\ \text{kg}$ bag weigh $5\ \text{kg}$ together, whichever way you hold them.

Vectors: a size and a direction

A vector has a magnitude and a direction. Displacement, velocity, acceleration, force and momentum are vectors.

Direction changes how vectors add. Two pulls of $30\ \text{N}$ and $40\ \text{N}$ on a box can give anything from $10\ \text{N}$ to $70\ \text{N}$. It depends on the angle between them. In the next lesson we learn the rule for adding them.

How we write a vector. We put an arrow over the letter, $\vec{A}$. Books also use bold type, A. The magnitude is written $|\vec{A}|$, or just $A$.

How we draw a vector. We draw an arrow. Its length shows the magnitude, and its head shows the direction. The end where the arrow starts is the tail.

Two vectors are equal when they have the same magnitude and the same direction. Where they start does not matter. Three cars each driving north at $60\ \text{km/h}$ on three different roads have equal velocities. So we may slide an arrow around the page, as long as we do not turn it or stretch it.

Is it a vector? Ask two questions

1. Does it have a direction?
2. Does it add like arrows, by the triangle rule?

A vector must pass both tests. Electric current in a wire passes the first, because it flows one way along the wire. It fails the second. Two currents of $2\ \text{A}$ and $3\ \text{A}$ that meet at a junction always give $5\ \text{A}$, whatever the angle between the wires. So current is a scalar.

Multiplying a vector by a number

What is $2\vec{A}$? It is the same arrow, twice as long. What is $-\vec{A}$? It is the same length, pointing the opposite way.

The rule, for a number $\lambda$ (the Greek letter lambda):

• If $\lambda$ is positive, $\lambda\vec{A}$ points the same way as $\vec{A}$, and its size is $\lambda A$.
• If $\lambda$ is negative, $\lambda\vec{A}$ points the opposite way, and its size is $|\lambda| A$.
• If $\lambda = 0$, we get the zero vector, $\vec{0}$. It has no length, and no direction.

Try it with the sliders. Watch the size readout as the arrow flips.

A number with a unit can turn one kind of vector into another. Divide a displacement by a time, and you get a velocity: $\vec{v} = \dfrac{\Delta\vec{r}}{\Delta t}$. Multiply an acceleration by a mass, and you get a force.

Unit vectors: direction only

Sometimes we want to say "this way" and nothing else. A unit vector does that. Its size is exactly $1$, and it has no unit. We write it with a hat: $\hat{A}$ (read "A hat").

To get the unit vector along $\vec{A}$, divide the vector by its own size:

Turned round, this says $\vec{A} = A\,\hat{A}$: any vector is its size times its direction. For example, a velocity of $15\ \text{m/s}$ towards the east is $15\ \text{m/s}$ times the unit vector that points east.

Position and displacement

Pick a fixed point O, the origin. The position vector $\vec{r}$ of a point P is the arrow from O to P. It tells you where P is.

When a body moves from P to Q, its displacement is the arrow from P to Q. It is the change in position: $\Delta\vec{r} = \vec{r}_Q - \vec{r}_P$. The displacement only cares about the start and the end. The winding path in between does not matter.

Watch out. These three points trip students up in exams.

• The magnitude of a vector is never negative. A minus sign in front of a vector only turns it round.
• The distance travelled is never less than the size of the displacement. They are equal only for a straight path with no turning back.
• A displacement can be zero after a long trip. Run one full lap of a track, and you end where you started.

Practise with new numbers

Each time you press New numbers, you get a new trip. Try a few until you can do it without the hint.

Try it in Python

The program works out the distance and the displacement for the walk at the start of this lesson.

Remember these

• A scalar has a size only. A vector has a size and a direction, and adds by the triangle rule.
• $\lambda\vec{A}$ points the same way as $\vec{A}$ if $\lambda > 0$, and the opposite way if $\lambda < 0$. Its size is $|\lambda|A$.
• A unit vector has size $1$: $\hat{A} = \vec{A}/A$, and $\vec{A} = A\hat{A}$.
• Displacement joins the start to the end. Distance is the length of the whole path.

Check yourself

Try each one on your own first. If you are stuck, open a hint. When you have an answer, check it with Show the answer.

1. Which of these are vectors: mass, force, speed, velocity, temperature, momentum, electric current?

2. A vector $\vec{A}$ has a magnitude of $6$. What are the magnitude and the direction of $-\frac{1}{2}\vec{A}$?

3. A runner completes one full lap of a $400\ \text{m}$ track. What is the distance she runs, and what is her displacement?

4. Can the size of a displacement ever be more than the distance travelled?