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Scalars and Vectors

Mathematics and Science developed to describe and understand our world and everything in our world.

We inhabit a world that has essentially three dimensions – a line, a flat shape, and a shape that has volume.

There is a fourth dimension and that is spacetime. These are regulated by time and three space dimensions – up & down, back & forth, and left & right.


To begin to understand scalars and vectors in mathematics, you have to know what the word magnitude means. Magnitude is a way of describing something in terms of size or weight and is expressed as a number.

For example, a brick has a magnitude of 15 lbs; we are describing its weight and / or how big it is. Comparisons can be made between two things that have magnitude.

weight diagram

A scalar is a quantity described by just the magnitude and, as mentioned, is just one single number. Speed, volume, weight, power, energy and time are all scalars.

Measurements like 40 mph, 20 cubic feet, 100 pounds (lbs), 20 hp and 5 hours are all expressions of scalars, and they all have just one number to describe them.

We don’t know anything else about what the number is describing.


For scalars, we can describe one thing as having more weight than another, greater speed than another or larger size than another. We can use words like smaller, larger, faster, or heavier, etc.

A vector describes a magnitude and a direction. Vectors are important when we want to describe motion.

Some examples of vectors include force, speed, acceleration, and momentum, direction, and distance.


So, a vector shows direction and a magnitude, and a scalar only has a magnitude. The easiest way to recognize a vector is that it includes a direction.

Let’s pause for some questions to check your understanding.


Which of the following involve vectors?

  1. A hockey player skating at 15 mph towards the goal.
  2. A box on the floor has a volume of 10 cubic feet.
  3. The temperature outside is 15 °C.
  4. A car is speeding south along a highway at 75 mph.


If you answered 1 and 4, you are right! Number 2 and 3 are scalars.

When working on any mathematical problem, such as adding, subtracting, multiplying and dividing, it involves only scalar numbers.

However, when solving problems involving vectors, it is much more complicated because you have to consider direction as well as the magnitude of the object.

old plane

Think of an airplane and all the forces that are involved with keeping the plane in the air. The scalar forces are:

Air density

Air pressure

Air temperature

Weight of the plane

Energy and work of the engine, without describing the direction.

Some of these forces combined with direction – such as thrust, weight moving forward, velocity, speed and acceleration – are vector quantities.


Quiz Time!

  1. How is a vector different from a scalar?
  2. What is magnitude?
  3. What is a scalar?
  4. Is wind travelling eastward at 30 mph and example of a vector or scalar?
  5. A plane parked on the runway is an example of vector or scalar?



  1. A vector has direction.
  2. Magnitude is way of describing something using size or weight.
  3. A scalar is a way of describing one magnitude.
  4. Wind travelling at 30 mph eastward is a vector.
  5. It is scalar because it is not moving in any direction.


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