Example

Suppose we know that (a, b) = 3 and (a, c) = 5.

Then we can deduce that (a, b + 2c) = 3 + 2 * 5 = 13.

Similarly, if you know the value of (a, b) and (a, c) you can deduce the value of the dot product of a with any vector that can be expressed as a linear combination of b and c, that is, as xb + yc.

The value will be x * (a, b) + y * (a, c).