Scalars and vectors

Size alone is not enough

Narrated video. Transcript:

Your school bag has a mass of 5 kg. The classroom is at 30 °C. The period lasts 40 minutes. Each of these is fully described by one number and a unit.

Quantities like these are called scalars. They have only a magnitude, and you combine them with ordinary arithmetic, like 5 kg plus 2 kg is 7 kg.

Now a friend says: walk 50 m from the school gate. Where do you end up? Watch the arrow spin around the gate. You could be anywhere on this circle.

You need a direction too: 50 m towards the north. A quantity that has both magnitude and direction, and adds by the triangle law, is called a vector.

Displacement, velocity, acceleration and force are vectors. Distance, mass, time and temperature are scalars.

We draw a vector as an arrow. Its length shows the magnitude, and the way it points shows the direction.

In print a vector is written in bold, like bold v. By hand, put a small arrow over the letter. Its magnitude is written |v|, or just v in italics.

So remember: magnitude alone makes a scalar. Magnitude plus direction makes a vector.

Position, displacement and equal vectors

Narrated video. Transcript:

Mark an origin O on the classroom floor and draw an x axis and a y axis through it.

A ball sits at point P. The arrow from O to P is the position vector of the ball, written r. It tells you how far P is from O, and in which direction.

Now the ball rolls to a new point, P′. Its new position vector is r′.

The arrow from P to P′ is the displacement. It joins the start to the end in a straight line, whatever route the ball took.

Watch three routes from P to P′: a straight one, a curved one and a zigzag. Their path lengths are different, but the displacement arrow is the same for all three.

Two vectors are equal if they have the same magnitude and the same direction. Slide B parallel to itself until its tail sits on the tail of A. If the tips meet too, A equals B.

These two arrows are the same length but point different ways. Their magnitudes are equal, but the vectors are not.

Vectors that we can slide parallel to themselves without changing them are called free vectors. Most vectors in physics are treated this way.

Multiplying a vector by a number

Narrated video. Transcript:

Take a vector A that points east and is 3 units long.

Multiply it by 2. The new arrow, 2A, points the same way and is twice as long: 6 units.

Multiply by one half instead. A/2 still points east, but it is only 1.5 units long.

Now multiply by −1. The length stays 3 units, but the arrow flips to point west. This is −A, the negative of A.

In general, λA has magnitude |λ| times A. It points along A when λ is positive, and opposite to A when λ is negative.

The number can carry units too. A velocity of 5 m/s east multiplied by a time of 10 s gives a displacement of 50 m east.

And if you multiply by zero, the arrow shrinks to nothing. That is the null vector, which we will meet again when we add vectors.

Try it: stretch and flip a vector

Interactive simulation. Things to try:

• Drag the blue tip to change A. The amber arrow is λA.
• Move the λ slider below zero and watch λA turn around.
• Check the readout: the size of λA is always |λ| times the size of A.

Quick check

Which one of these is a vector quantity?

1. Speed
2. Mass
3. Displacement
4. Distance travelled

Negative multiples

A vector A has magnitude 4 m and points north-east. What is −1.5 A?

1. 6 m towards the north-east
2. 6 m towards the south-west
3. −6 m towards the north-east
4. 2.5 m towards the south-west

Summary

• Position vector of P: r = OP (from origin O to P)
• Displacement from P to P′: PP′ = r′ − r
• Multiplying by a number: |λA| = |λ| A

• A scalar has only magnitude. A vector has magnitude and direction and adds by the triangle law.
• Displacement joins start to end in a straight line. Its magnitude is never more than the path length.
• Two vectors are equal only if both their magnitude and direction are the same.
• λA is |λ| times as long as A. It points along A if λ > 0 and opposite to A if λ < 0.