What would be the sum of two dot products with a common vector?

Question

Let's assume there are 3 vectors, $\vec x$, $\vec y$ and $\vec z$. Is there a way to simplify $$(\vec x\cdot\vec y + \vec x\cdot\vec z)$$ or do I need to operate both the dot products before adding them? Thanks in advance!

Answer 1

The dot product is distributive over vector addition.

For example, if $\vec x=(x_1,x_2), \vec y=(y_1,y_2)$, and $\vec z=(z_1,z_2)\in\mathbb R^2$, then

$\vec x\cdot\vec y +\vec x\cdot \vec z=x_1y_1+x_2y_2+x_1z_1+x_2z_2=x_1(y_1+z_1)+x_2(y_2+z_2)=\vec x\cdot(\vec y+\vec z)$.