Dot or scalar product | Improve Society

The dot or scalar product of two vectors $\bold{A}$ and $\bold{B}$ , denoted by $\bold{A}\cdot\bold{B}$ (read $\bold{A}$ dot $\bold{B}$ ) is defined as the product of the magnitude of $\bold{A}$ and $\bold{B}$ and the cosine of the angle between them. In symbols,
$\bold{A}\cdot\bold{B} = AB\cos\theta,\ 0\le\theta\le\pi\cdots(4)$
Note that $\bold{A}\cdot\bold{B}$ is a scalar and not a vector.

The following laws are valid:

$\bold{A}\cdot\bold{B} = \bold{B}\cdot\bold{A}$

Communicative Law for Dot Products

$\bold{A}\cdot(\bold{B} + \bold{C}) = \bold{A}\cdot\bold{B} + \bold{A}\cdot\bold{C}$

Distributive Law

$m(\bold{A}\cdot\bold{B}) = (m\bold{A})\cdot\bold{B} = \bold{A}\cdot(m\bold{\bold{B}}) = (\bold{A}\cdot\bold{B})m$

where $m$ is a scalar.

$\bold{i}\cdot\bold{i} = \bold{j}\cdot\bold{j} = \bold{k}\cdot\bold{k} = 1,\ \bold{i}\cdot\bold{j} = \bold{j}\cdot\bold{k} = \bold{k}\cdot\bold{i} = 0$

If $\bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k}$ and $\bold{B} = B_1\bold{i} + B_2\bold{j} + B_3\bold{k}$ then

$\bold{A}\cdot\bold{B} = A_1B_1 + A_2B_2 + A_3B_3$

$\bold{A}\cdot\bold{A} = A^2 = A_1^2 + A_2^2 + A_3^2$

$\bold{B}\cdot\bold{B} = B^2 = B_1^2 + B_2^2 + B_3^2$

If $\bold{A}\cdot\bold{B} = 0$ and $\bold{A}$ and $\bold{B}$ are not null vectors, then $\bold{A}$ and $\bold{B}$ are perpendicular.