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What is the difference between a vector and a scalar, and how do we combine quantities that have direction?

Vectors and scalars: distinguishing the two kinds of quantity, the difference between distance and displacement and between speed and velocity, and combining vectors that act at right angles.

An SQA National 5 Physics answer on vectors and scalars, covering which quantities are scalar and which are vector, the difference between distance and displacement and between speed and velocity, and how to combine two vectors that act at right angles using a scale diagram or Pythagoras.

Generated by Claude Opus 4.89 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this key area is asking
  2. Scalars and vectors
  3. Distance and displacement, speed and velocity
  4. Combining vectors at right angles
  5. Why direction matters
  6. Try this

What this key area is asking

The SQA wants you to know which physical quantities are scalars and which are vectors, to tell the difference between distance and displacement and between speed and velocity, and to combine two vectors that act at right angles to find a resultant.

Scalars and vectors

Knowing which list a quantity belongs to is worth easy marks and stops mistakes later. For example, weight is a force and so a vector that always points downwards, while mass is a scalar that has no direction.

Distance and displacement, speed and velocity

The speed and velocity relationships look almost the same but use different quantities:

Average speed uses the total distance, so it is a scalar. Average velocity uses the displacement, so it is a vector and must be given with a direction. A car driving a winding road can have a high average speed but a small average velocity if it ends up close to where it started.

Combining vectors at right angles

When two vectors act at right angles (for example a journey east then north, or two forces at 90∘90^{\circ}), the resultant is the diagonal of the rectangle they form. You can find it in two ways, and the SQA accepts both:

  • Scale diagram. Draw each vector tip to tail to scale, using a ruler and protractor, then measure the resultant arrow and its angle.
  • Calculation. Use Pythagoras for the size and the tangent ratio for the direction.

Why direction matters

Because velocity is a vector, a change of direction is a change of velocity even if the speed stays the same. A car going round a roundabout at a steady 30 km/h30 \text{ km/h} is changing velocity all the time, which is why it needs a force (friction from the tyres) to turn. This idea returns when you study acceleration and Newton's laws.

Try this

Q1. State whether each quantity is a scalar or a vector: mass, force, distance, velocity. [2 marks]

  • Cue. Mass scalar, force vector, distance scalar, velocity vector.

Q2. A runner completes one full lap of a 400 m400 \text{ m} track. State the distance run and the displacement. [2 marks]

  • Cue. Distance 400 m400 \text{ m}; displacement 0 m0 \text{ m} (finishes at the start).

Q3. Two forces of 3.0 N3.0 \text{ N} and 4.0 N4.0 \text{ N} act at right angles. Calculate the size of the resultant. [2 marks]

  • Cue. R=3.02+4.02=25=5.0 NR = \sqrt{3.0^2 + 4.0^2} = \sqrt{25} = 5.0 \text{ N}.

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SQA N5 style4 marksA hill walker walks 3.0 km due east and then 4.0 km due north. Determine the total distance walked and the magnitude and direction of the displacement.
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The total distance is a scalar, so add the two legs: 3.0+4.0=7.0 km3.0 + 4.0 = 7.0 \text{ km}.

The displacement is a vector, found by combining the two legs at right angles. Use Pythagoras: s=(3.0)2+(4.0)2=9.0+16.0=25=5.0 kms = \sqrt{(3.0)^2 + (4.0)^2} = \sqrt{9.0 + 16.0} = \sqrt{25} = 5.0 \text{ km}.

The direction is the angle from east: θ=tan⁡−1 ⁣(4.03.0)=53∘\theta = \tan^{-1}\!\left(\frac{4.0}{3.0}\right) = 53^{\circ}, so the displacement is 5.0 km5.0 \text{ km} at a bearing of 037037 (that is, 53∘53^{\circ} north of east).

Markers reward the scalar sum for distance, the use of Pythagoras for the magnitude, and a stated direction for the displacement.

SQA N5 style2 marksState the difference between a scalar quantity and a vector quantity, and give one example of each.
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A scalar quantity has magnitude (size) only. An example is distance, speed, mass, energy or time.

A vector quantity has both magnitude and direction. An example is displacement, velocity, acceleration, force or weight.

Markers reward the magnitude-only and magnitude-and-direction descriptions and one correct example of each.

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