In mathematics and physics, a scalar is a quantity that only has magnitude (size), while a vector has both magnitude and direction. Examples of scalar quantities include pure numbers, mass, speed, temperature, energy, volume, and time. Examples of vector quantities include velocity, acceleration, momentum, displacement, and forces, such as weight and friction.
|Indicates magnitude or size||Indicates magnitude and direction|
|Examples include mass, temperature, speed, and time||Examples include velocity, acceleration, weight, and friction|
|May or may not have units (e.g., degrees, kilograms)||May be written in bold text or with an arrow over it to distinguish from scalar. Direction may be positive/negative or other (e.g., east, left, down).|
|Mathematical operations involving scalars produce another scalar. Operations on a scalar and a vector produce a vector.||Operations involving vectors produce a scalar or a vector. The dot product of two vectors is a scalar. Summation, subtraction, or the cross product of two vectors is a vector.|
Examples of Scalars
Here are some examples of scalar quantities:
Examples of Vectors
Here are some examples of vector quantities:
Scalar vs Vector – Test Your Understanding
(1) The car is going 75 mph.
This is a scalar value because you don’t know which direction the car is going.
(2) You walked 4 mph toward the store.
This is a vector because you have both a magnitude and a direction.
(3) The box in the west corner of the room has a mass of 12 kilograms.
The mass of the box is a scalar quantity. Even though you know the location of the box, this fact has nothing to do with its mass.
(5) The time is 12:30 pm.
This is a scalar. There is no direction.
(6) The pressure inside a balloon is 2 atmospheres.
Pressure has a magnitude, but it does not have a direction. Another way of looking at it is that pressure acts in all directions at once.
(7) The cat weighs 8 pounds.
Mass and weight can be confusing, when it comes to distinguishing between scalar and vector quantities. The pound is a unit of weight, so this value is a vector. The implied direction is down, toward the Earth’s center of gravity. If the “weight” of the cat was given in kilograms, it would be a scalar value (for mass, not weight). A cat’s weight is different on the Moon or Mars, but its mass remains the same.
- A unit vector is a vector that has a magnitude of 1. Usually, it’s indicated by placing a carat (^) over it. The unit vector x, with a carat over it, is read as “x-hat” because the vector looks somewhat like it’s wearing a hat.
- The null vector or zero vector is a vector with a magnitude of zero. While it has no magnitude, it has a direction. For example, you could use a null vector to describe which direction a compass is pointing.
- Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. Toronto: Thomson Learning. ISBN 978-0-03-083993-1.
- Banach, Stefan (1922). “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)”. Fundamenta Mathematicae (in French). 3: 133–181. doi:10.4064/fm-3-1-133-181
- Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4.
- Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6.