Scalars and vectors

AS Level Physics · Physical quantities and units. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

A scalar has magnitude only. A vector has magnitude and direction. Scalars add like ordinary numbers, but vectors must be added with their directions taken into account.

To add two vectors, draw them head to tail. The resultant is the line from the start of the first to the end of the second. For vectors at right angles use Pythagoras. To subtract a vector, reverse its direction and then add. This is how you find a change in velocity: final velocity minus initial velocity.

Any vector can be replaced by two perpendicular components that together have the same effect. Resolving into components turns a problem with angles into two simple problems.

Rules to remember

Common mistake

Using cos for the horizontal component every time. Cos goes with the component next to the given angle, so check whether the angle is measured from the horizontal or from the vertical.

Worked example

A box of weight 50 N rests on a slope at 20° to the horizontal. What are the components of its weight along the slope and at right angles to the slope? (sin 20° = 0.342, cos 20° = 0.940)

  1. The angle between the weight and the line at right angles to the slope is 20°, the same as the angle of the slope.
  2. Component at right angles to the slope (next to the angle) = 50 × cos 20° = 50 × 0.940 = 47 N.
  3. Component along the slope (opposite the angle) = 50 × sin 20° = 50 × 0.342 = 17.1 N.

Answer: 17 N down the slope and 47 N at right angles to the slope.

Practice questions

Try each one, then open the answer.

1. Two forces, each of 10 N, act at a point with an angle of 120° between them. What is the magnitude of the resultant force?

  1. A
    0 N
  2. B
    10 N
  3. C
    17 N
  4. D
    20 N
Show answer

Answer: B. Drawn head to tail, the two forces and the resultant form an equilateral triangle. The resultant is 2 × 10 cos 60° = 10 N, along the bisector.

2. A ball is kicked at 15 m s−1 at 40° above the horizontal. What is the vertical component of its initial velocity? (sin 40° = 0.643, cos 40° = 0.766)

  1. A
    9.6 m s−1
  2. B
    11.5 m s−1
  3. C
    12.6 m s−1
  4. D
    15 m s−1
Show answer

Answer: A. The vertical component is opposite the 40° angle: 15 sin 40° = 15 × 0.643 = 9.6 m s−1. 11.5 m s−1 is the horizontal component.

3. Which list contains only vector quantities?

  1. A
    force, velocity, momentum
  2. B
    mass, weight, force
  3. C
    speed, displacement, acceleration
  4. D
    energy, power, momentum
Show answer

Answer: A. Force, velocity and momentum all have direction. Mass, speed, energy and power are scalars.

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