Scalars and vectors: (a) scalar and vector quantities

The idea

A scalar quantity has magnitude only, while a vector quantity has both magnitude and direction. Mass, time, temperature, speed, distance, energy and power are scalars. Displacement, velocity, acceleration, force, momentum, weight and electric field strength are vectors. Direction matters because vector quantities combine by direction as well as size: walking 3 m east then 4 m north gives a displacement of 5 m on a bearing, not 7 m. When solving problems, first classify each quantity, then decide whether to add magnitudes directly for scalars or resolve and combine directions for vectors. In equations, a scalar such as mass multiplies a vector such as velocity to give the vector momentum, so the direction is retained.

What earns the marks

• A scalar has magnitude only; examples include mass, time, temperature, speed, distance, energy and power.
• A vector has magnitude and direction; examples include displacement, velocity, acceleration, force, momentum and weight.
• Scalars combine by ordinary arithmetic addition of their magnitudes.
• Vectors must be combined using direction, for example by resolving components or drawing a vector triangle.
• Multiplying or dividing a vector by a scalar changes the magnitude but not the direction, unless the scalar is negative.
• Distance is a scalar while displacement is a vector; speed is a scalar while velocity is a vector.

Common mistakes and how to put them right

• Treating speed as a vector.

  Put it right: Speed has magnitude only, whereas velocity includes direction.

• Adding vector magnitudes arithmetically regardless of direction.

  Put it right: Two perpendicular 3 N and 4 N forces give a resultant of 5 N, not 7 N.

• Assuming distance and displacement are interchangeable.

  Put it right: Displacement is the straight-line distance in a specified direction.

• Believing that a negative sign on a scalar quantity indicates spatial direction.

  Put it right: For scalars like temperature or charge, a negative sign indicates a value below zero or a type of charge, not a physical direction.

Exam tips

• Underline the direction words in a question, such as north, upwards or at 30°, before choosing a method.
• When asked to classify quantities, give the reason: state whether direction is needed to define the quantity.
• Use standard symbols and units consistently, for example velocity $v$ in m s⁻¹ and displacement $s$ in m.