Question

find the value of x for which the vectors (3i-4j×xk) & (2i-j-3k) are orthogonal​

Answer 1

To FinD :-

• The value of x for which the two given vectors are orthogonal .

$\red{\frak{ Given}}\begin{cases} \sf Let \ \vec{A} = ( 2\hat{i} - 4\hat{j} + x\hat{k} ) \\\sf Let \ \vec{B} = ( 2\hat{i} - 1\hat{j} -3\hat{k} ) \end{cases}$

We need to find the value of x for which the two vectors are othogonal , that is perpendicular to each other. We know that for two vectors to be perpendicular their dot product should be 0 . So that ,

$\sf :\implies \vec{A} .\vec{B}= 0 \\\\\sf:\implies ( 2\hat{i} - 4\hat{j} + x\hat{k} ) . ( 2\hat{i} - 1\hat{j} -3\hat{k} ) = 0 \\\\\sf:\implies (2\hat{i} . 2\hat{i}) -4\hat{j}. -1\hat{j} + (x\hat{k} . -3\hat{k} ) = 0 \\\\\sf:\implies 4 +4 -3x = 0 \\\\\sf:\implies 8-3x = 0 \\\\\sf:\implies 3x = 8 \\\\\sf:\implies \underset{\blue{\sf Required\ Answer }}{\underbrace{\boxed{\pink{\frak{ x =\dfrac{8}{3}}}}}}$