Can a vector be distributed into a difference if it is being multiplied via scalar product?

Question

Let's say I have $(\vec{v_1}-\vec{v_2})\cdot \vec{v_3}$. Can I do this:

$$=(\vec{v_1}\cdot \vec{v_3}-\vec{v_2}\cdot \vec{v_3})$$

Answer 1

Yes you can. That is one of the scalar product's properties, bilinearity:

$$\langle \alpha v + \beta u \cdot w \rangle = \alpha \langle v\cdot w \rangle + \beta \langle u \cdot w \rangle $$

Alogside with commutativeness:

$$\langle v \cdot u\rangle = \langle u \cdot v\rangle$$

And a property whose name I know not:

$$\langle v \cdot v\rangle \ge 0$$ with $\langle v \cdot v\rangle = 0 \iff v = 0$

Answer 2

Let $v^1_k$, $v^2_k$ and $v^3_k$ be the $k$-th components of $v_1, v_2$ and $v_3$. Then the $k$-th component ot $v_1 - v_2$ is $v^1_k-v^2_k$. Therefore you have

$(v_1 - v_2) \cdot v_3 = \sum_{k=1}^N (v^1_k-v^2_k)v^3_k =$

$ =\left(\sum_{k=1}^N v^1_k v^3_k\right) - \left(\sum_{k=1}^N v^2_k v^3_k\right) = (v_1 \cdot v_3) - (v_2 \cdot v_3). $