General Physics> Sir Pls prove that the vector area of a t...

Sir
Pls prove that the vector area of a triangle whose vertices are a vector bvector and cvector is
0.5(bvec cross cvector +cvector cross a vec+ a vector cross b vector)
Thanks and Rgds,
Jai

Jai Mahajan , 11 Years ago

Grade 12

1 Answers

Sumit Majumdar

Last Activity: 11 Years ago

Dear student,

Let the vertices be given by the vectors:

$\overrightarrow{A}=\left ( a_{1}, a_{2}, a_{3} \right ), \overrightarrow{B}=\left ( b_{1}, b_{2}, b_{3} \right ), \overrightarrow{C}=\left ( c_{1}, c_{2}, c_{3} \right )$

So, the area would be given by:

$\Delta =\frac{1}{2}\sqrt{\begin{vmatrix} a_{2} & a_{3} & 1\\ b_{2} & b_{3} & 1\\ c_{2} & c_{3}& 1 \end{vmatrix}^{2}+\begin{vmatrix} a_{3} & a_{1} & 1\\ b_{3} & b_{1} & 1\\ c_{3} & c_{1}& 1 \end{vmatrix}^{2}+\begin{vmatrix} a_{1} & a_{2} & 1\\ b_{1} & b_{2} & 1\\ c_{1} & c_{2}& 1 \end{vmatrix}^{2}}$ $\Delta =\frac{1}{2}\sqrt{\begin{vmatrix} a_{2} & a_{3} & 1\\ b_{2} & b_{3} & 1\\ c_{2} & c_{3}& 1 \end{vmatrix}^{2}+\begin{vmatrix} a_{3} & a_{1} & 1\\ b_{3} & b_{1} & 1\\ c_{3} & c_{1}& 1 \end{vmatrix}^{2}+\begin{vmatrix} a_{1} & a_{2} & 1\\ b_{1} & b_{2} & 1\\ c_{1} & c_{2}& 1 \end{vmatrix}^{2}}=\frac{1}{2}\left | \left ( A\times B \right )\cdot C \right |$