Calculating with Vectors

Vectors help us describe motion and forces using size and direction together. For example, 5 m/s east is a velocity (vector). 5 m/s without a direction is speed (scalar).

Key ideas

• Scalar: size only (distance, speed, time, mass).
• Vector: size and direction (velocity, force, acceleration, weight).
• Resultant vector: the single vector that has the same effect as two or more vectors combined.

Adding vectors

Same line: treat one direction as positive. Example: 10 N right and 6 N left give 4 N right (10 − 6).

At right angles (perpendicular): use Pythagoras for the size and trigonometry for the direction.

Size (magnitude) of resultant, $R$, from perpendicular components $A$ and $B$:

$$R = \sqrt{A^2 + B^2}$$

Direction angle $\theta$ (measured from the axis you choose as the “adjacent”):

$$\theta = \tan^{-1}\!\left(\dfrac{\text{opposite}}{\text{adjacent}}\right)$$

Always include units with the magnitude and a clear direction (for example, “north of east”).

Graphical (scale diagram) method

• Choose a scale (for example, 1 cm = 1 m/s or 1 N).
• Draw the first vector as an arrow.
• From its tip, draw the second vector (tail-to-tip).
• Draw the resultant from the start of the first to the tip of the second. Measure its length and angle.

Common mistakes

• Adding sizes without considering direction.
• Forgetting to include the direction of the resultant.
• Using Pythagoras when vectors are not perpendicular.
• Mixing units (for example, cm with m).
• Measuring angle from the wrong axis or the wrong way round.