Question

show that the vector a vector =2i^-3j^+7k^ and b vector = 4i^ +6j^-14k^ are mutually parallel vector​

Answer 1

Given :

$:\implies\sf\:\overrightarrow{A}=2\hat{i}-3\hat{j}+7\hat{k}$

$:\implies\sf\:\overrightarrow{B}=4\hat{i}+6\hat{j}-14\hat{k}$

To Prove :

Both vectors are mutually parallel.

Explanation :

★ Cross product of two vectors is given by

$\dag\:\:\underline{\boxed{\bf{\orange{\overrightarrow{A}\times\overrightarrow{B}=AB\:sin\theta}}}}$

For parallel vectors, A × B = 0

$\sf\:\overrightarrow{A}\times\overrightarrow{B}=\begin{array}{|ccc|}\sf\hat{i}&\sf\hat{j}&\sf\hat{k}\\ \sf A_x&\sf A_y&\sf A_z\\ \sf B_x&\sf B_y&\sf B_z\end{array}$

$\sf\:\overrightarrow{A}\times\overrightarrow{B}=(A_yB_z-A_zB_y)\hat{i}-(A_xB_z-B_xA_z)\hat{j}+(A_xB_y-A_yB_x)\hat{k}$

Cross product of both vectors must be zero if they are mutually parallel.

★ Cross Product :

$\sf\:[(-3)(-14)-(7)(6)]\hat{i}-[(2)(-14)-(7)(4)]\hat{j}+[(2)(6)-(-3)(4)]\hat{k}$

$\sf\:[42-42]\hat{i}-[-28+28]\hat{j}+[12-12]\hat{k}$

$\bf\:=0$

Hence proved!!