Math Formulas

Scalar Product Formulas

Scalar Product is also known as Dot Product.

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

no-image
$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$
${\theta}=\;cos^{-1}{\left(\dfrac{b}{a}\right)}$

Where ${\theta}$ is the angle between vector a and vector b.

Scalar 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}$

$\vec{a} \cdot \vec{b} = a_1b_1+a_2b_2+a_3b_3$
Angle between two vectors :

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

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

$cos\theta=$ $\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}$
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.

Orthogonal Vectors :

Two vectors are $\vec {a}$, $\vec {b}$ are orthogonal or perpendicular if $\vec {a} \cdot \vec {b} =0$.

If $\vec {a} \cdot \vec {b} =0$
Where, $\theta=\dfrac{\pi}{2}$
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

$\vec {i} \cdot \vec {i}=\vec {j} \cdot \vec {j}=\vec {k} \cdot \vec {k}=1$

$\vec {i} \cdot \vec {i}=\vec {j} \cdot \vec {j}=\vec {k} \cdot \vec {k}=1$