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Module 2: Foundations in physics›Scalars and vectors (2.3.1)

Scalars and vectors (2.3.1)

Adding and resolving vectors, components, vector triangles, parallelogram rule, and distinguishing scalar from vector quantities in A-level Physics.
6 min

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Scalar quantities have magnitude but no direction.

A table titled 'Scalar quantities' with two columns. The first column lists various scalar quantities: Distance, Speed, Mass, Time, Temperature, Energy, Power, Density, Work done, and Pressure. The second column provides descriptions for each quantity: Distance - How far an object travels (no direction), Speed - How fast an object moves, regardless of direction, Mass - Amount of matter in an object, Time - Duration of an event, Temperature - Measure of thermal energy or particle motion, Energy - Capacity to do work, Power - Rate of energy transfer, Density - Mass per unit volume, Work done - Energy transferred when a force moves an object, Pressure - Force per unit area (often treated as scalar at GCSE). The table is attributed to Medify.

An example of a scalar quantity is mass. The average mass of a human is which is just a number and has no direction.

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Vector quantities have both magnitude and direction.

Vector quantities and their descriptions: Displacement - The straight-line change in position, including both direction and magnitude. Velocity - Speed in a particular direction. Weight - Force due to gravity acting on a mass (acts towards the centre of Earth). Acceleration - Rate of change of velocity, including direction. Force - A push or pull acting in a specific direction. Momentum - Product of mass and velocity, direction same as velocity. Impulse - Change in momentum caused by a force acting over time. Drag / Frictional force - Resistive force acting opposite to motion. Lift - Upward force on an object in a fluid, opposite to weight. Magnetic field strength (field lines) - Has both magnitude and direction around magnets or currents.

An example of a vector quantity is force. A person of mass standing on the surface of Earth feels a force due to their weight with magnitude (where is the gravitational field strength) and direction pointing towards the Earth’s centre.

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Scalar quantities with the same units can be added or subtracted from each other.

For example, two rulers placed end to end have a total length:

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A vector can be represented visually by an arrow:

  • The length of the arrow is proportional to the vector magnitude.
  • The arrow points in the direction of the vector.
  • A vector is usually written with an arrow above the letter:
  • A vector can also be represented by an underlined letter:
A grid with horizontal and vertical lines. A green line extends horizontally from a green dot on the left to an arrow pointing right.

The diagram above shows a force vector drawn on paper with squares:

  • If the scale is then the force has a magnitude
  • The direction of the arrow indicates that the force is directed to the right.
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The diagram below shows the addition of two vectors and

  • Vectors can be added by placing the arrows end to end, as shown below.
  • The resultant vector can be found by drawing an arrow from the start of the first vector arrow to the end of the second arrow.
A graph with a grid background showing three vectors: a red vector labeled v→r, a green vector labeled v→1, and another green vector labeled v→2. The vectors originate from a green point at the bottom left corner of the graph.
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The diagram below shows the subtraction of two vectors from

  • To subtract one vector from another, treat it as adding the negative: . This means reversing the direction of to get , then adding it to end-to-end.
  • The resultant vector runs from the tail of the first vector to the tip of the reversed vector . It represents the difference between the two original vectors, both in magnitude and direction
A graph with a grid background showing two vectors. The first vector, represented by an arrow labeled →v1, is green and moves horizontally to the right. The second vector, represented by an arrow labeled →vr, is red and moves diagonally downward to the left. There is also an arrow labeled −→v2, which is green and moves vertically downward.
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The resultant of any two coplanar vectors can be determined by a scale drawing.

The addition of two coplanar displacement vectors is drawn in the example below:

  • For a scale, the magnitude of the resultant vector can be measured by a ruler as
  • The angle of the resultant vector to the horizontal is measured as
A graph showing vectors with labels: v1, v2, and vr. The angle θ is indicated at the origin where the vectors originate. The grid background is marked with horizontal and vertical lines.
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The magnitude of the resultant vector, can be found from Pythagoras’ theorem:

Where and are the magnitudes of the two perpendicular vectors.

The diagram below shows the vector addition of two perpendicular displacement vectors.

A graph showing vectors in a coordinate system. The red vector is labeled with v_r, and it points diagonally upwards. The green vector is labeled with v_1 and v_2, pointing horizontally and vertically, respectively. An angle θ is indicated at the origin.

In the diagram above, and so:

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Trigonometric relationships can be used to determine the direction of a resultant vector that is formed by two vectors and acting perpendicularly to one another.

For the diagram below, the angle, of the resultant vector to the horizontal can be found from:

A graph showing two vectors. The red vector labeled v_r points diagonally, with v_2 = 5 m at the top right and v_1 = 3 m at the bottom left. An angle θ is indicated at the origin.

In the diagram above is opposite the angle while is adjacent. Therefore, the angle can be calculated as follows:

It is useful to note that the magnitude and direction of any resultant vector can also be calculated for any coplanar vector using the cosine rule and the sine rule.

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The sine rule is a remarkable mathematical fact that allows you to find unknown angles and lengths in triangles that do not have a right angle.

This rule is particularly useful when combining two vectors that are not perpendicular to each other, and either the value of one of the vectors or the angle of one of the vectors is missing.

A triangle labeled with angles and sides: angle A is 27°, angle B is 39°, angle C is 114°, side a is 8, side b is 11, side c is 16. The equations shown are sin 27° / 8 = 0.057, sin 39° / 11 = 0.057, sin 114° / 16 = 0.057.

The sine rule is often written as:

Where:

  • is the side opposite angle
  • is the side opposite angle and
  • is the side opposite angle

It is important to note that the sine rule does not have to be memorised. If an exam question requires its use, then the formula will be provided in your formula booklet.

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The cosine rule is a formula that can be used to calculate a missing side or angle in a triangle.

This formula is particularly useful when combining two vectors that are not perpendicular to each other, and either the value of one of the vectors or the angle of one of the vectors is missing.

A triangle labeled with points A, B, and C. The angle at point A is shaded pink and labeled with the letter 'b'. The side opposite angle A is labeled 'a' in red, while the side opposite point B is labeled 'c'. The title 'COSINE RULE' is displayed at the top.

An example of the cosine rule is illustrated in the image above: the side a can be found using the formula:

If the angle is missing, then this can be found using the cosine rule rearranged as below:

It is important to note that the cosine rule does not have to be memorised. If an exam question requires its use, then the formula will be provided in your formula booklet.

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Question walkthrough

Finding an Angle Using the Sine and Cosine Rules

Uses the sine and cosine rules to find an unknown angle in a vector triangle formed by combining two vectors of given magnitude and direction.

A vector can be resolved into its perpendicular components.

A force acting in the plane may be resolved into its and components. For a force with magnitude pointing at an angle to the X axis:

  • The horizontal component magnitude is
  • The vertical component magnitude is
A graph showing a vector in a Cartesian coordinate system. The vector is represented by a red arrow labeled F with its magnitude |F| = F. The angle θ is indicated at the base of the vector. The horizontal component of the vector is labeled Fx = F cos θ, and the vertical component is labeled Fy = F sin θ. The y-axis is vertical and the x-axis is horizontal.

There are many contexts where resolving a vector into its perpendicular components is useful, often when an object is constrained to move in one direction or when only one direction of the vector is relevant:

  • An example of this is in foot races, the wind velocity component parallel to the track must be calculated to determine the headwind or tailwind during a race.
  • Another example of this is projectile motion, in which an object is acted on by gravity so that its horizontal velocity component remains the same (ignoring air resistance) but its vertical velocity component varies.
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