What is the difference between scalar values and vectors?

What is a scalar?

Stated simply a scalar has a magnitude, only. Typically scalar values are real numbers. For example, the number four is a scalar value, where four is the magnitude.

What is a vector?

A vector has a magnitude like a scalar, but it also has a direction. For example, a vector might be expressed as <2, 3>. This notation breaks down the vector into its components. Another notation for vectors is polar notation. An example of polar notation is 4 at angle 45 degrees. This notation identifies the magnitude and the direction.

In practice, both scalars and vectors typically also are expressed with units. For example, saying the length of something is four is meaningless without units. Saying something is 4 meters in length makes the value usable.

Converting vectors to scalars

Sometimes we are not concerned with the direction of a vector, and simply want to know it's magnitude. In this case, we can take magnitude of the vector. In polar notation getting a magnitude of a vector is relatively simple. In the example above of four at angle 45°, the magnitude is four.

In the other vector example above, <2, 3>, to get the magnitude, we can employ trigonometry. The two magnitudes two and three can be envisioned as forming the adjacent at opposite sides of a triangle. Then the vector's magnitude is the length of the hypotenuse of the triangle. So, by the Pythagorean theorem, we know that the link to the hypotenuse would be sqrt(2*2 + 3*3) or sqrt(4+9) or sqrt(13).