Math Formulas
Triple Product
Scalar Triple Product :
- $\Big[\vec{a}~~\vec{b}~~\vec{c}\Big]=\vec{a} \cdot \Big(\vec{b} \times \vec{c}\Big)$
- $\Big[\vec{a}~~\vec{b}~~\vec{c}\Big]=\Big[\vec{c}~~\vec{a}~~\vec{b}\Big]$ $=\Big[\vec{b}~~\vec{c}~~\vec{a}\Big]$ $= -\Big[\vec{b}~~\vec{a}~~\vec{c}\Big]$ $=-\Big[\vec{c}~~\vec{b}~~\vec{a}\Big]$ $=-\Big[\vec{a}~~\vec{c}~~\vec{b}\Big]$
- $\lambda \vec{a} \cdot \Big(\vec{b} \times \vec{c}\Big) =\lambda \Big[\vec{a}~~\vec{b}~~\vec{c}\Big] $
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{c} = (c_1,c_2,c_3) = c_1\vec{i} + c_2\vec{j}+c_3\vec{k}$
$\begin{align*}
\vec{a} \cdot \left(\vec{b} \times \vec{c}\right) =
\left|
\begin{array}{ccc}
a_1 & a_2 & a_3\\
b_1 & b_2 & b_3\\
c_1 & c_2 & c_3
\end{array}
\right|
\end{align*}$
- If $\vec{a} \cdot \left(\vec{b} \times \vec{c}\right)=0~~$, then the $\vec{a}, \vec{b}~ \text{and} ~ \vec{c}~$ are linearly dependent. So, $\vec{c}=\lambda ~\vec{a} + \mu ~\vec{b}~ $ for some scalars $ \lambda ~ \text{and} ~ \mu$.
- If $\vec{a} \cdot \left(\vec{b} \times \vec{c}\right)≠0~$, then the $\vec{a},\vec{b}~ \text{and} ~\vec{c}~$ are linearly independent.
Volume of parallelepiped :
Volume = Area of base ⋅ height
Volume = $ \Big| \left(\vec{a} \times \vec{b}\right) \cdot \vec{c} ~\Big|$
Volume = $ \Big| \left(\vec{a} \times \vec{b}\right) \cdot \vec{c} ~\Big|$
Vector Triple Product :
$\vec{a} \times \Big(\vec{b} \times \vec{c}\Big) = \Big(\vec{a} \cdot
\vec{c}\Big)\vec{b}$ $-\Big(\vec{a} \cdot \vec{b}\Big)\vec{c}$