Geometrical Interpretation

Projections. If $\vec a$ is any vector and $\hat n$ a unit vector, then $\vec a\cdot\hat n$ is the (signed) projection of $\vec a$ onto the line along $\hat n$: it is positive when the angle between $\vec a$ and $\hat n$ is acute, and negative when that angle is obtuse.

For arbitrary non-zero vectors $\vec a,\vec b$: $\left|\dfrac{\vec a\cdot\vec b}{|\vec b|}\right|=\left|\dfrac{\vec a\cdot\vec b}{|\vec a|}\right|$ can be read as the length of the projection of $\vec b$ along $\vec a$'s direction or of $\vec a$ along $\vec b$'s direction. We recall $\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta$, where $\theta$ is the angle from $\vec a$ to $\vec b$ measured counter-clockwise.

Remark 6.1(1). The angle between two non-zero vectors is $\theta=\cos^{-1}\left(\dfrac{\vec a\cdot\vec b}{|\vec a||\vec b|}\right)$.

Remark (2). $\vec a,\vec b$ are parallel iff the angle between them is $0$ or $\pi$.

Remark (3). $\vec a,\vec b$ are perpendicular iff the angle between them is $\pi/2$ or $3\pi/2$.

Properties (for any non-zero $\vec a,\vec b$):

For any vectors $\vec a,\vec b,\vec c$ and scalar $\alpha$: $\vec a\cdot\vec b=\vec b\cdot\vec a$ (commutative), $\vec a\cdot(\vec b+\vec c)=\vec a\cdot\vec b+\vec a\cdot\vec c$ (distributive), $\vec a\cdot(\alpha\vec b)=\alpha(\vec a\cdot\vec b)=(\alpha\vec a)\cdot\vec b$; and $\vec a\times\vec b=-(\vec b\times\vec a)$ (anticommutative, not commutative), $\vec a\times(\vec b+\vec c)=\vec a\times\vec b+\vec a\times\vec c$, $\vec a\times(\alpha\vec b)=\alpha(\vec a\times\vec b)=(\alpha\vec a)\times\vec b$. …