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Two vector quantities can be combined into a scalar quantity because a vector lives in a vector space, which requires the existence of an operation called the dot product (also commonly known as the scalar product or inner product).

The exact form of this operation depends on the type of vector space, and of course one can define other operations which map two vectors into a scalar.

A commonly used definition is as follows:

Imagine vector one contains these values (x1, x2, x3, x4) and vector two contains these values (y1, y2, y3, y4), the dot product would turn this into:

x1*y1 + x2*y2 + x3*y3 + x4*y4

The dot product gives a measure of the angle between two vectors and is often used as such in for example mechanics.

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