Scalars, vectors, resolving and resultants

Scalars and vectors

• A scalar has size (magnitude) only: mass, distance, speed, time, energy, work, power, density.
• A vector has size and direction: displacement, velocity, acceleration, force, weight, momentum.

Vectors are printed in bold (F) or drawn as arrows whose length shows the size.

Resolving a vector

A vector F at angle θ to the horizontal has two perpendicular components:

• horizontal: F cos θ
• vertical: F sin θ

The component next to the angle uses cos. You can also resolve by scale drawing.

Adding two vectors

• At right angles: resultant = sqrt(A² + B²), and the angle from A is tan^-1(B / A).
• At any angle: draw them to scale, tip to tail (or as a parallelogram), and measure the resultant with a ruler and protractor.

The size of the resultant of two vectors A and B always lies between A - B and A + B (where A is the larger).

Worked example

A force of 50 N acts at 30° above the horizontal. Find its horizontal and vertical components.

1. Horizontal = 50 cos 30° = 43.3 N
2. Vertical = 50 sin 30° = 25 N

Answer: 43 N horizontally and 25 N vertically

Check yourself

1. Which of these is a vector?
   • Displacement
   • Speed
   • Mass
   • Energy
   Show answer

   Displacement. Displacement has a direction; the others have magnitude only.

2. Which of these is a scalar?
   • Momentum
   • Weight
   • Work done
   • Acceleration
   Show answer

   Work done. Work done (energy) has no direction.

3. A force of 20 N acts at 60° to the horizontal. Find its horizontal component in N.
   Show answer

   10 N. 20 cos 60° = 20 x 0.5 = 10 N.

4. A ball is kicked at 12 m/s at 35° above the horizontal. Find the vertical component of its velocity in m/s, to 3 s.f.
   Show answer

   6.88 m/s. 12 sin 35° = 6.88 m/s.

5. Forces of 6.0 N east and 8.0 N north act on a point. Find the size of the resultant force in N.
   Show answer

   10 N. sqrt(6.0² + 8.0²) = sqrt(100) = 10 N.

6. For the forces in the last question (6.0 N east and 8.0 N north), find the angle between the resultant and east, in degrees, to 3 s.f.
   Show answer

   53.1 °. tan θ = 8.0 / 6.0, so θ = 53.1°.

7. Forces of 5 N and 3 N act on an object at any angle to each other. Which resultant is NOT possible?
   • 2 N
   • 9 N
   • 5 N
   • 8 N
   Show answer

   9 N. The resultant must lie between 5 - 3 = 2 N and 5 + 3 = 8 N.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)

Last updated 7 November 2026. Spotted a mistake? Tell us.