#### Answer

$\vec{A}\cdot \vec{B}=A_xB_x + A_yB_y+A_zC_z$

#### Work Step by Step

a) Anything dotted with itself is one, so the value of all of these dot products (also known as scalar products) is one.
b) We know that anything dotted with a different unit vector is 0, so the value of all of these dot products is 0.
c) Using the distributive property and the results from a) and b), we find:
$\vec{A}\cdot \vec{B}=A_xB_x + A_yB_y+A_zC_z$