Vectors and Scalars Explained
Vectors and scalars explained: the difference, examples of each, adding vectors with scale drawings and Pythagoras, resolving components and worked examples.
Key takeaways
- A scalar has size only (such as mass or speed); a vector has size and direction (such as force or velocity).
- Add vectors tip to tail; for two at right angles, use Pythagoras for the size and trigonometry for the direction.
- Any vector can be split into perpendicular components, such as F cos θ and F sin θ.
On this page
In physics, every quantity is either a scalar or a vector. A scalar has size (magnitude) only, such as a mass of 5 kg or a speed of 20 m/s. A vector has both size and direction, such as a force of 10 N downwards or a velocity of 20 m/s north. The difference matters because vectors combine differently: walking 3 km east and then 4 km north covers 7 km of distance, but leaves you only 5 km from where you started.
Examples of scalars and vectors
| Scalars (size only) | Vectors (size and direction) |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Mass | Weight (a force) |
| Time | Acceleration |
| Energy | Force |
| Temperature | Momentum |
| Power | Electric field |
Some pairs look similar but aren’t the same. Distance is how far you travel; displacement is how far you end up from the start, in a particular direction. Speed is how fast; velocity is how fast in a given direction. Our guide to speed, velocity and acceleration explains these in more detail.
Writing vectors
Vectors are drawn as arrows: the length shows the size, and the arrow points in the direction. In print they’re often written in bold (F) or with an arrow above the symbol, and directions are given as compass points, bearings or angles from a reference line.
Adding vectors in a straight line
If vectors act along the same line, add them, treating one direction as positive:
- A 10 N force right plus a 4 N force right gives 14 N right.
- A 10 N force right plus a 4 N force left gives 6 N right.
When forces balance, the resultant is zero, and the object stays still or keeps moving at a steady velocity, as described by Newton’s first law.
Adding vectors at right angles
Place the vectors tip to tail. The resultant goes from the start of the first to the tip of the last. For two vectors at right angles:
- Size: use Pythagoras. R = √(a² + b²).
- Direction: use trigonometry. tan θ = opposite ÷ adjacent.
Example: a walk. You walk 3 km east, then 4 km north. The displacement is √(3² + 4²) = 5 km. The angle north of east is tan⁻¹(4 ÷ 3) ≈ 53.1°, a bearing of about 037°.
Example: crossing a river. A boat heads straight across a river at 4 m/s while the current flows at 3 m/s. The resultant velocity is √(4² + 3²) = 5 m/s, at tan⁻¹(3 ÷ 4) ≈ 36.9° downstream from straight across.
Scale drawings
For vectors at any angle, a scale drawing works well:
- Choose a scale, such as 1 cm = 1 N.
- Draw the first vector to scale in the right direction.
- Draw the second starting from the tip of the first.
- Draw the resultant from the start to the final tip.
- Measure its length and angle, then convert back using the scale.
Resolving vectors into components
Any vector can be split into two perpendicular parts, usually horizontal and vertical. For a vector of size F at an angle θ to the horizontal:
- Horizontal component = F × cos θ
- Vertical component = F × sin θ
Example: a child pulls a sledge with a 50 N force at 30° above the horizontal. The horizontal component, which moves the sledge forward, is 50 × cos 30° ≈ 43.3 N. The vertical component, 50 × sin 30° = 25 N, slightly lifts the sledge, reducing friction; see friction explained.
Resolving is how projectile motion is worked out: the horizontal and vertical parts of a launch velocity are treated separately.
Resultant forces and free-body diagrams
A free-body diagram shows all the forces acting on one object as arrows. Adding them gives the resultant force, which determines acceleration through F = m × a. On a slope, for example, weight is resolved into a component down the slope (m g sin θ) and one into the slope (m g cos θ).
| Situation | Forces | Resultant |
|---|---|---|
| Book resting on a table | Weight down, reaction up, equal | Zero; the book stays still |
| Car accelerating | Driving force forward larger than drag and friction | Forward; the car speeds up |
| Skydiver at terminal velocity | Weight down equals air resistance up | Zero; steady speed |
Vectors in everyday life
- Navigation: pilots and sailors combine their own velocity with wind or current.
- Sport: the angle of a kick splits the ball’s velocity into height and distance.
- Engineering: forces in bridges and cranes are resolved into components.
- Momentum: collisions are worked out with vectors because direction matters; see momentum explained.
- Gravity: the force of gravity always points towards the centre of the attracting mass.
Common mistakes
- Adding vector sizes directly when they point in different directions.
- Mixing distance and displacement, or speed and velocity.
- Using sin and cos the wrong way round. Cos goes with the side next to the angle.
- Forgetting direction in the final answer. A vector answer needs a size and a direction.
Frequently asked questions
Is time a vector or a scalar?
A scalar. Time has size but no direction in everyday physics.
Is weight a vector?
Yes. Weight is a force, so it has both size and direction, always towards the centre of the Earth (or another planet).
Can a vector have a negative value?
Yes. A negative sign shows the vector points in the opposite direction to the one chosen as positive.
Sources
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