Direct comparison
Scalar vs vector
A scalar quantity has magnitude only; a vector quantity has both magnitude and a direction.
The step most authors miss
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Side-by-side comparison
| Dimension | Scalar | Vector |
|---|---|---|
| What it has | Magnitude (size) only. | Magnitude and direction. |
| Direction? | No direction associated. | Direction is an essential part of it. |
| Examples | Mass, temperature, time, speed, energy, distance. | Displacement, velocity, acceleration, force, momentum. |
| How they add | Add by ordinary arithmetic. | Add geometrically, taking direction into account. |
| Can be negative? | Usually only as a value below a reference (e.g. temperature). | A sign or angle indicates direction. |
| Specified by | A single number and a unit. | A number, a unit and a direction. |
| Notation | A plain symbol, e.g. m for mass. | Bold or an arrow, e.g. v with an arrow over it. |
| Speed vs velocity | Speed is the scalar. | Velocity is the vector. |
| Distance vs displacement | Distance is the scalar. | Displacement is the vector. |
Why vectors need special arithmetic
Because vectors carry direction, you cannot simply add their sizes. Two forces of 10 newtons add to 20 newtons only if they point the same way; pointing in opposite directions they cancel to zero, and at right angles they combine to about 14 newtons. This geometric addition is why physics distinguishes scalars from vectors so carefully: getting the direction right is as important as getting the size right when predicting motion, equilibrium or the net effect of several influences.
Common questions
FAQ
Is speed a scalar or a vector?+
Speed is a scalar — it tells you how fast something moves but not in which direction. Velocity is the corresponding vector, giving both the speed and the direction of motion. The same pattern appears with distance (scalar) and displacement (vector).
How do you tell a scalar from a vector?+
Ask whether direction is part of the quantity. Mass, time, temperature, speed and energy have no direction, so they are scalars. Force, velocity, acceleration and displacement only make sense with a direction attached, so they are vectors. A vector needs a magnitude, a unit and a direction to be fully specified.
Why can vectors not just be added like ordinary numbers?+
Because their direction affects the result. Two equal vectors pointing the same way add to double the size, but pointing in opposite directions they cancel, and at an angle they combine to something in between. Vector addition uses geometry — drawing them head to tail or resolving into components — rather than simple arithmetic.
Going deeper
