Question

How do you find the cross product and dot product $?$

Answer 1

Finding dot product:

The dot product of two vectors $\alpha$ and $\beta$ can be found using the formula

$\overrightarrow{\mathbf{A}}\mathbf{}\mathbf{·}\mathbf{}\overrightarrow{\mathbf{B}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathit{A}\mathbf{\right|}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{\left|}\mathit{B}\mathbf{\right|}\mathbf{}\mathbf{\times }\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{\theta }\mathbf{\right)}$

Where

$\left|A\right|$ is the magnitude of vector $A$

$\left|B\right|$ is the magnitude of vector $B$

$\theta$ is the angle between vector $A$ and $B$.

Finding cross product:

The cross product of two vectors $\alpha$ and $\beta$ can be found using the formula

$\mathit{A}\mathbf{}\mathbf{\times }\mathbf{}\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathit{A}\mathbf{\right|}\mathbf{}\mathbf{\left|}\mathit{B}\mathbf{\right|}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathit{\theta }\mathbf{\right)}\mathbf{}\mathit{n}$

Where

$\left|A\right|$ is the magnitude of vector $A$

$\left|B\right|$ is the magnitude of vector $B$

$\theta$ is the angle between vector $A$ and $B$.

$n$ is the unit vector which is at the right angle to both the vectors.

Hence, we can find the cross product by the formula $\mathit{A}\mathbf{}\mathbf{\times }\mathbf{}\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathit{A}\mathbf{\right|}\mathbf{}\mathbf{\left|}\mathit{B}\mathbf{\right|}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathit{\theta }\mathbf{\right)}\mathbf{}\mathit{n}$ and dot product by the formula $\mathit{A}\mathbf{}\mathbf{·}\mathbf{}\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathit{A}\mathbf{\right|}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{\left|}\mathit{B}\mathbf{\right|}\mathbf{}\mathbf{\times }\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{\theta }\mathbf{\right)}$.