Let $\mathbf{a},\mathbf{b} \in \mathbb{R^3}$. Let $a_1,a_2,a_3,b_1,b_2,b_3 \in \mathbb{R}$ such that
$\mathbf{a} = (a_1,a_2,a_3)$ and $\mathbf{b} = (b_1,b_2,b_3)$. Let $\hat{i}$, $\hat{j}$ and $\hat{k}$ denote unit vectors along the positive $x$,$y$ and $z$ axis of a right handed coordinate system.
Definition: The cross product of $\mathbf{a}$ and $\mathbf{b} $, denoted by $\mathbf{a} \times \mathbf{b}$ is defined as:-
$$\mathbf{a} \times \mathbf{b} := (a_2b_3 – a_3b_2)\hat{i} + (a_3b_1 – a_1b_3)\hat{j} + (a_1b_2 – a_2b_1)\hat{k}$$
Using this definition of $\mathbf{a} \times \mathbf{b}$, I wanted to show the following property:
$$\mathbf{a} \times \mathbf{b} = |\mathbf{a} | |\mathbf{b} | \sin (\theta) \hat{n} $$
where $\theta$ is the non-negative non-reflex angle between $\mathbf{a}$ and $\mathbf{b}$ and $\hat{n}$ is a unit vector perpendicular to $\mathbf{a}$ and $\mathbf{b}$ whose direction is given by the right hand thumb rule.
Right-Hand Thumb Rule for $\mathbf{a} \times \mathbf{b} $: If the index finger of your right hand is along the direction of $\mathbf{a}$, and your middle finger is along the direction of $\mathbf{b}$, then the thumb gives the direction of $\mathbf{a} \times \mathbf{b} $:
I was able to show that $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a} | |\mathbf{b} | \sin (\theta) $ and $\mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) = 0$ and $\mathbf{b} \cdot (\mathbf{a} \times \mathbf{b}) = 0$ using plain algebra. Hence I have shown that $\hat{n}$ is perpendicular to $\mathbf{a}$ and $\mathbf{b}$. However, I don't know how to show that the direction of $\hat{n} $ by the right hand thumb rule is the same as the direction of $\mathbf{a} \times \mathbf{b} $ as given by the definition of cross product. How can I show this?
Best Answer
To prove that the Right-Hand Thumb Rule holds we can start by the definition for the unitary vectors along coordinates axes
$\vec i \times \vec j=\vec k, \qquad \vec j \times \vec k=\vec i, \qquad \vec k\times \vec i=\vec j$
$\vec j \times \vec i=-\vec k, \qquad \vec k \times \vec j=-\vec i, \qquad \vec i\times \vec k=-\vec j$
and assume wlog