Inner products and distributive property

Question

Is this true for the inner products ? :
$(\vec a + \vec b)\cdot(\vec c + \vec d) = \vec a\cdot\vec c + \vec a\cdot\vec d + \vec b\cdot\vec c + \vec b\cdot\vec d$.

Answer 1

It is indeed true. It should be easy to confirm this by writing out the vectors explicitly in component form and evaluating both sides of the equality.

Remember that if $\mathbf{x} = \langle x_1, x_2, ..., x_n \rangle$ and $\mathbf{y} = \langle y_1, y_2, ..., y_n \rangle$, then:

$$\mathbf{x} \cdot \mathbf{y} = \sum_{k=1}^n x_ky_k$$