Extended · Practice questions · Resultant of perpendicular vectors

Pythagoras, then the angle.

Six original Cambridge-style Extended questions on finding a resultant by calculation and by scale drawing, and on the two traps: adding perpendicular vectors directly, and forgetting the direction.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method to apply

Square, add, root; then the tangent.

Magnitude: R = √(x² + y²) for two perpendicular vectors.
Direction: θ = tan⁻¹(y ÷ x), and state which reference it is measured from.
Always give direction: a resultant is a vector, so size alone is incomplete.

Calculation

[2 marks]

Two forces act on a point at right angles: 3.0 N pointing east and 4.0 N pointing north. Calculate the magnitude of the resultant force.

Pythagoras.

R = √(3.0² + 4.0²) = √(9 + 16) = √25

R = 5.0 N

Calculation

[3 marks]

A horizontal force of 12 N and a vertical force of 5.0 N act on an object at right angles. Find the magnitude and direction of the resultant.

Magnitude.

R = √(12² + 5.0²) = √(144 + 25) = √169

R = 13 N

Direction.

θ = tan⁻¹(5.0 ÷ 12) = 23°

above the horizontal (12 N) force. ✓

Analysis

[2 marks]

A student adds a 3 N force and a 4 N force that act at right angles and writes "resultant = 7 N." Explain why this is wrong and state the correct magnitude.

The forces are perpendicular, so they combine by Pythagoras, not by simple addition. ✓
R = √(3² + 4²) = 5 N. ✓ (adding to 7 N would only be right if they pointed the same way)

Analysis

[3 marks]

Describe how to find the resultant of two perpendicular forces using an accurate scale drawing.

Choose a scale (for example 1 cm to 1 N) and draw the two forces accurately at right angles. ✓
Complete the rectangle and draw the diagonal from the starting point. ✓
Measure the diagonal's length (convert back using the scale) for the magnitude, and measure its angle with a protractor for the direction. ✓

Calculation

[4 marks]

A boat's engine drives it east at 8.0 m/s. A current carries it north at 6.0 m/s. Find the magnitude and direction of the boat's resultant velocity.

Magnitude.

v = √(8.0² + 6.0²) = √(64 + 36) = √100

v = 10 m/s

Direction.

θ = tan⁻¹(6.0 ÷ 8.0) = 37°

north of east. So 10 m/s at 37° north of east. ✓

Analysis

[2 marks]

A question asks for the resultant of two perpendicular forces. A student correctly calculates 10 N but writes only "10 N" as the final answer. State what is missing and why it costs a mark.

The direction is missing (for example the angle to one of the forces). ✓
A resultant is a vector, so it is only fully described by both a magnitude and a direction. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.