Scalar Product

Definition. Let $\vec a,\vec b$ be two non-zero vectors with included angle $\theta$. Their scalar product (or dot product), written $\vec a\cdot\vec b$, is defined as the number $\vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\theta.$ Because the result of $\cdot$ is a scalar, this is called the scalar product; because the symbol used is a dot, it's also called the dot product.

Geometric meaning — projection. Let $\vec{OA}=\vec a,\ \vec{OB}=\vec b$, and drop a perpendicular from $B$ to line $OA$, meeting it at $L$. In right triangle $OLB$, $\cos\theta=\dfrac{OL}{OB}$, so $OL=|\vec b|\cos\theta$ — and $OL$ is exactly the projection of $\vec b$ onto $\vec a$ (the 'shadow' $\vec b$ casts along the direction of $\vec a$). Then $\vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\theta=|\vec a|\,(OL)=|\vec a|\times(\text{projection of }\vec b\text{ on }\vec a).$ Rearranging, **projection of …