2.1 Scalars and vectors — Learn

Prerequisites: SI units and prefixes from Section 1.1; standard form; the idea that a quantity is a number with a unit; the compass directions north, south, east and west.

By the end of this section you can: tell a scalar quantity from a vector quantity; write a vector as a magnitude and a direction; use positive and negative signs for direction along a line; add vectors that act along one line; and multiply a vector by a scalar.

1. Two kinds of quantity

A scalar has a magnitude only. Mass, time, temperature and energy are scalars. A mass of 5 kg and a temperature of 20 °C need no direction to make sense.

A vector has a magnitude and a direction. Displacement, velocity, acceleration and force are vectors. A force of 20 N is not a complete statement. The force of 20 N pushing north is a complete statement.

The magnitude of a vector is its size, written without a direction. The magnitude of a velocity is a speed. The magnitude of a displacement is a distance.

2. A direction is not enough on its own

Some scalar quantities have a direction in everyday language. Electric current flows along a wire, and a temperature can be negative. Neither is a vector.

The test is how the quantities combine. Two vectors add by the triangle rule. Place the tail of the second vector at the tip of the first. Draw the resultant from the tail of the first to the tip of the second. A quantity is a vector only if it combines in that way.

Two currents do not combine by the triangle rule. A current of 3 A and a current of 4 A meeting at a junction give a current of 7 A, not 5 A. So current is a scalar, even though it flows in a direction.

Watch out. A sign is not the same as a direction. A temperature of $-5\,^\circ\text{C}$ is not a vector, and neither is a bank balance of $-\pounds 20$. A vector needs a direction in space, and it must add by the triangle rule. Check each quantity by asking: if I put two of these end to end, does the triangle rule give the combined effect?

3. Try it: adding two vectors

The figure places two forces tip to tail and draws the resultant. Three sliders set the two magnitudes and the angle between them.

Predict first. Do not move the sliders yet. Write down your answers.

1. Force A is $6\,\text{N}$ and force B is $8\,\text{N}$. The angle between them is $90^\circ$. Is the resultant $2\,\text{N}$, $10\,\text{N}$ or $14\,\text{N}$?
2. What happens to the resultant when the two forces point in opposite directions?
3. The forces have magnitudes $6\,\text{N}$ and $8\,\text{N}$. Can the resultant ever be longer than $14\,\text{N}$?

Test. Set A to $6\,\text{N}$, B to $8\,\text{N}$ and the angle to $90^\circ$, then read the resultant. Move the angle to $0^\circ$ and then to $180^\circ$, and watch the resultant. Check your second and third predictions.

Explain. Why is the resultant shorter than the two magnitudes added when the forces point in different directions? Write two or three sentences. Use the idea that the second force starts at the tip of the first.

4. Writing a vector

A vector is written as a magnitude and a direction. The direction can be given in several ways.

• A bearing, measured clockwise from north. A displacement of 250 m on a bearing of 060° points 60° clockwise from north, which is roughly north-east.
• An angle from a stated direction. A force of 50 N at 30° above the horizontal.
• A pair of components. A displacement of 3.0 m east and 4.0 m north can be written as the vector $(3.0, 4.0)\,\text{m}$. You will use components in Section 2.2.

In printed work a vector is often shown in bold, such as $\mathbf{v}$, or with an arrow, such as $\vec{v}$. Its magnitude is written $v$ or $|\mathbf{v}|$. In handwriting an underline, $\underline{v}$, is common.

5. Vectors along one line

When two vectors act along the same line, choose one direction as positive. A vector in the opposite direction then has a negative sign. Adding the signed numbers gives the resultant, and its sign gives the direction.

Worked example 1. A walker walks 3.0 km east, then 1.5 km west. Find the resultant displacement.

The resultant is 1.5 km east. The distance travelled is 4.5 km, but the displacement is only 1.5 km.

Worked example 2. A cyclist travels 500 m east in 20 s. Find the average velocity.

The velocity is $25\,\text{m}\,\text{s}^{-1}$ east. The speed is $25\,\text{m}\,\text{s}^{-1}$, with no direction stated.

Worked example 3. A boat has a speed through the water of 6.0 m s⁻¹ east. The current is 2.0 m s⁻¹ east. Find the velocity over the ground when the boat travels with the current and against it.

The two velocities are along the same line, so the signed numbers add. With the current the two speeds add, and against it they subtract.

6. Multiplying a vector by a scalar

Multiplying a vector by a positive scalar changes its magnitude and leaves its direction unchanged. Multiplying by a negative scalar reverses the direction.

Worked example 4. A force of 20 N acts east. Find the force 3 times larger in the same direction.

Multiplying by $-1$ would give 20 N west. Doubling a displacement doubles its length but keeps it pointing the same way.

7. Speed and velocity round a bend

A car on a roundabout can keep a steady speed while its velocity changes. The speed is the magnitude of the velocity. The direction of the velocity changes as the car turns, so the velocity vector changes even when its length does not.

On a straight runway the direction of motion stays the same, so one velocity describes the whole run. On a curved road the direction keeps changing, so the velocity keeps changing. Distance and speed describe how much ground is covered. Displacement and velocity describe the change in position, and they carry a direction.

Progress check

Return to the chapter problem. The ferry moves at 4.0 m s⁻¹ through the water and the current flows at 3.0 m s⁻¹.

1. Classify the ferry's speed through the water and its velocity over the ground as scalar or vector.
2. The ferry points straight across. Is the current along the same line as the ferry's motion, or at right angles to it?
3. Say whether the two velocities can be added as ordinary numbers.

If question 1 goes wrong, revisit Section 1 and check whether a direction is stated. If question 2 goes wrong, draw the two directions as arrows from a single point.

You should now be able to

• tell a scalar from a vector, and give an example of each;
• explain why a stated direction alone does not make a quantity a vector;
• write a vector as a magnitude and a direction, including as a bearing;
• add and subtract vectors that act along one line;
• multiply a vector by a scalar, and state the effect on the magnitude and the direction.