Scalars and vectors: direction is the dividing line
Definition
Scalar quantity
A quantity that has magnitude (size) only.
Definition
Vector quantity
A quantity that has both magnitude (size) and direction.
- Both kinds of quantity have a number and a unit; the difference is whether a direction is needed as well.
- "A mass of 5 kg5\ \text{kg}5 kg" is complete on its own, but "a force of 5 N5\ \text{N}5 N" is not, because 5 N5\ \text{N}5 N upwards and 5 N5\ \text{N}5 N downwards do opposite things to the same object.
- This matters because vectors do not add up the way ordinary numbers do.
- Two forces of 3 N3\ \text{N}3 N give 6 N6\ \text{N}6 N in the same direction, 0 N0\ \text{N}0 N in opposite directions, and about 4.2 N4.2\ \text{N}4.2 N at right angles, whereas two masses of 3 kg3\ \text{kg}3 kg always give 6 kg6\ \text{kg}6 kg.
The scalars and vectors you need to recognise
- Scalars: distance, speed, time, mass, energy, temperature.
- Vectors: force, weight, velocity, acceleration, momentum, electric field strength, gravitational field strength.
- Speed (scalar) contrasts with velocity (vector, speed in a stated direction), so a car circling a roundabout at a steady 30 km/h30\ \text{km/h}30 km/h has constant speed but changing velocity.
- Mass (scalar, kg\text{kg}kg) contrasts with weight (vector, N\text{N}N), because weight is the gravitational force acting vertically downwards.
Key Idea
- If a quantity is a force of any kind, weight, friction, drag, or an electric or magnetic force, it is a vector.
- If it is an amount of "stuff", such as mass, energy, time or temperature, it is a scalar.
- Field strengths are vectors too: gravitational field strength, 9.8 N/kg9.8\ \text{N/kg}9.8 N/kg near the Earth's surface, points towards the centre of the Earth.
Finding the resultant of two vectors at right angles
Definition
Resultant vector
The single vector that has the same effect as two or more vectors acting together.
- This is limited to two vectors at right angles, and to forces or velocities.
- Because the two vectors are perpendicular, they form a right-angled triangle with the resultant as the hypotenuse.
- You may be asked to work it out by calculation or graphically, so you need both methods.
By calculation
- Find the magnitude from Pythagoras' theorem: R=A2+B2R = \sqrt{A^2 + B^2}R=A2+B2.
- Find the direction from the tangent ratio, as an angle measured from one of the original vectors: tanθ=BA\tan\theta = \dfrac{B}{A}tanθ=AB, so θ=tan−1 (BA)\theta = \tan^{-1}\!\left(\dfrac{B}{A}\right)θ=tan−1(AB).
Graphically (scale drawing)
- Choose a scale and write it down, for example 1 cm=1 N1\ \text{cm} = 1\ \text{N}1 cm=1 N.
- Draw the first vector to scale in the correct direction, with an arrowhead.
- From the tip of the first vector, draw the second to scale at 90∘90^\circ90∘ to it, tip-to-tail.
- Close the triangle from the start of the first vector to the tip of the second: that line is the resultant.
- Measure its length and convert with your scale, then measure its angle with a protractor.
Common Mistake
- Do not add magnitudes arithmetically when vectors are at right angles: 12 N12\ \text{N}12 N and 5.0 N5.0\ \text{N}5.0 N give 13 N13\ \text{N}13 N, never 17 N17\ \text{N}17 N.
- Mass is not weight: mass (kg\text{kg}kg) is a scalar that does not change with location, while weight (N\text{N}N) is a vector force that changes when ggg changes.
- Speed and velocity are not interchangeable: something moving in a circle at constant speed has a changing velocity, and therefore an acceleration, because its direction changes.
- A negative sign on a vector means "in the opposite direction", not "smaller": a deceleration of −2 m/s2-2\ \text{m/s}^2−2 m/s2 is an acceleration directed backwards.
- Energy is a scalar, so never describe a force in joules.
Exam technique
- Draw the right-angled triangle first, even when you are told to calculate; it stops you mixing up which side is the hypotenuse.
- Quote the direction the way the question frames it, for example "23∘23^\circ23∘ east of north"; a bare angle with no reference direction loses the direction mark.
- For a scale drawing, write the scale on the page and keep the precision sensible, since a ruler and protractor cannot justify three significant figures.
- The final answer mark needs the unit: N\text{N}N for a force, m/s\text{m/s}m/s for a velocity.
Self review
- What is the difference between a scalar and a vector quantity?
- Give three scalar quantities and three vector quantities.
- How do you find the magnitude and direction of the resultant of two perpendicular vectors?
- Why does a car going round a roundabout at constant speed still have an acceleration?
- What must you always state alongside the angle when giving the direction of a resultant?
