Math Formulas || Vectors || Vector Product

Math Formulas

Vector Product Formulas

Vector Product is also known as Cross Product.

Vector Product of $\vec{a}~ \text{ and }~ \vec{b} $

$\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta$

Where, $0 ≤{\theta} ≤ \dfrac{\pi}{2}$

Vector Product in Co-ordinate Form :

$\vec{a} = (a_1,a_2,a_3) = a_1\vec{i} + a_2\vec{j}+a_3\vec{k}$

$ \vec{b} = (b_1,b_2,b_3) = b_1\vec{i} + b_2\vec{j}+b_3\vec{k}$

$\vec{a} \times \vec{b} = a_1b_2 \vec{k}$ $ - a_1b_3 \vec{j}$ $- a_2b_1 \vec{k}$ $~ + a_2b_3 \vec{i}$ $ + a_3b_1 \vec{j}$ $ - a_3b_2 \vec{i}$

$\Rightarrow \vec{a} \times \vec{b} =$ $(a_2b_3-a_3b_2 )\vec{i}$ $~ - (a_1b_3-a_3b_1) \vec{j}$ $ +(a_1b_2-a_2b_1) \vec{k}$

$\begin{align*} \vec{a} \times \vec{b} = \left| \begin{array}{ccc} \vec{i} & \vec{j} & \vec{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{array} \right| \end{align*}$

Angle between two vectors :

$sin\theta=\dfrac{\vec{a} \times \vec{b}}{|\vec{a}| |\vec{b}|}$

Properties of Dot Product :

1. Commutative Property :

For any two vectors $\vec{a}$ and $\vec{b}$,

$\vec{a} \cdot \vec{b} =\vec{b} \cdot \vec{a}$

2. Associative Property :

For any three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$,

$\vec {a} \cdot \left(\vec{b} + \vec{c} \right)=\vec {a} \cdot \vec {b} +\vec {a} \cdot \vec {c} $

3. Distributive Property :

$(\lambda\vec{ a})(\mu \vec{ b})=\lambda \mu ~\vec {a} \cdot \vec {b} $

Where, $\lambda$ is a scalar quantity.

Parallelsim :

Two vectors are $\vec {a}$, $\vec {b}$ are parrallel if $\vec {a} \times \vec {b} =0$.

If $\vec {a} \times\vec {b} =0$
then, $\theta=0$

Standard Unit Vector :

If $ ~ \vec{a} = (a_1,a_2,a_3) = a_1\vec{i} + a_2\vec{j}+a_3\vec{k} ~$, then

$\vec {i} \times \vec {i}=\vec {j} \times \vec {j}=\vec {k} \times \vec {k}=0 $

$\vec {i} \times \vec {j}=\vec {k}$
$\vec {j} \times \vec {k}=\vec {i}$
$\vec {k} \times \vec {i}=\vec {j}$

NOTE :

$\vec{a} \times \vec{b}$ is a vector that is perpendicular to both $ \vec{a}$ and $ \vec{b}$.
The magnitude (or length) of the vector $\vec{a} \times \vec{b}$, written as $| \vec{a} \times \vec{b}|$, is the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$
The direction of $\vec{a} \times \vec{b}$ is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from $\vec{a}$ to $\vec{b}$, then the thumb points in the direction of $\vec{a} \times \vec{b}$.)