Question

What is the relation between dot product and cross product?

Answer 1

Step 1: Assumptions and known relations:

1. Consider two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ and let the angle between the two vectors be $\theta$.
2. The dot product of the two vectors is $\overrightarrow{a}·\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta ...\left(1\right)$.
3. The magnitude of the cross product of the two vectors is $\left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta ...\left(2\right)$

Step 2: Finding the relationship between dot and cross product:

Squaring both sides of equation $\left(1\right)$, we get, ${\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2{\cos }^2\theta ...\left(3\right)$

Similarly, squaring both sides of equation $\left(2\right)$, we get, ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2{\sin }^2\theta ...\left(4\right)$

Adding equations $\left(3\right)$ and $\left(4\right)$,

$\begin{array}{rcl}{\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2& =& {\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2{\cos }^2\theta +{\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2{\sin }^2\theta \\ {\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2& =& {\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2\left({\cos }^2\theta +{\sin }^2\theta \right)\\ {\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2& =& {\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2-{\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2\end{array}$

Hence, the relationship between dot and cross product is $\begin{array}{rcl}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2& =& {\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2-{\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2\end{array}$.