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Physical quantities & measurement

Resultant of two vectors

Resultant of perpendicular vectors Supplement 3 min read

Extended (Supplement)

The resultant of two vectors

The resultant of two or more vectors is the single vector that has the same effect as all the original vectors combined. The process of finding it is called vector addition.

At IGCSE Extended, you only need to find the resultant of two vectors that act at right angles to each other (perpendicular). This is done either by calculation or by a scale drawing.

Vectors at right angles — Pythagoras and trigonometry

When two vectors are perpendicular, they form the two shorter sides of a right-angled triangle. The resultant is the hypotenuse.

Step-by-step method (calculation)
  1. Draw a rough sketch showing the two vectors at 90° to each other (tip-to-tail).
  2. Apply Pythagoras: \( R = \sqrt{a^2 + b^2} \)
  3. Find the angle: \( \theta = \arctan\!\left(\dfrac{b}{a}\right) \) — the angle between the resultant and the larger vector.
  4. State the resultant as: magnitude + angle from a reference direction.
a b R θ
Worked example — two forces at right angles

A boat experiences a force of 30 N due east from its engine and a current force of 40 N due north. Find the resultant force (magnitude and direction).

Step 1 — Magnitude:

\( R = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ N} \)

Step 2 — Direction:

\( \theta = \arctan\!\left(\dfrac{40}{30}\right) = \arctan(1.33) \approx 53° \text{ from east} \) (i.e. N 53° E, or 53° north of east)

Answer: The resultant force is 50 N at 53° north of east.

Graphical method — scale drawing

An alternative to calculation: draw the vectors to scale, tip-to-tail, and measure the resultant directly.

Scale drawing method
  1. Choose a suitable scale (e.g. 1 cm : 10 N).
  2. Draw the first vector as an arrow to scale in the correct direction.
  3. From the tip of the first vector, draw the second vector to scale in its direction.
  4. Draw the resultant arrow from the tail of the first vector to the tip of the second.
  5. Measure the length of the resultant arrow and convert using the scale. Measure the angle with a protractor.
When using a scale drawing, always state your scale clearly (e.g. "Scale: 1 cm = 5 N") and show the measurement of the resultant arrow and the measured angle. The examiner cannot award marks for measurements they cannot verify.

Resultant of two velocities

The same method applies to velocity vectors. A classic scenario is a boat crossing a river.

Worked example — river crossing

A swimmer can swim at 2 m/s perpendicular to a riverbank. The river current flows at 1.5 m/s parallel to the bank. Find the resultant velocity of the swimmer.

\( R = \sqrt{2^2 + 1.5^2} = \sqrt{4 + 2.25} = \sqrt{6.25} = 2.5 \text{ m/s} \)

\( \theta = \arctan\!\left(\dfrac{1.5}{2}\right) = \arctan(0.75) \approx 37° \) from the direction of swimming (i.e. angled 37° downstream).

Answer: Resultant velocity = 2.5 m/s at 37° downstream from the perpendicular to the bank.

The examiners use the 3–4–5 right-angled triangle frequently for resultant vector calculations (30 N and 40 N → 50 N; 6 m/s and 8 m/s → 10 m/s). Recognising these Pythagorean triples saves time and avoids calculator errors.

Equilibrium and zero resultant

If two equal and opposite forces act on an object, their resultant is zero — the object is in equilibrium (no net force, so no acceleration). Equilibrium is explored further in the Forces chapter.