If vectors $\vec{A}=\hat{i}-3\hat{j}+5\hat{k}$ and $\vec{B}=7\hat{i}+2\hat{j}-x\hat{k}$ are perpendicular to each other, then the value of $x$ is

Solution:

Two vectors $\vec{A}$ and $\vec{B}$ are perpendicular to each other if their scalar (dot) product is zero, i.e., $\vec{A} \cdot \vec{B} = 0$.

Given vectors:

$\vec{A} = \hat{i} - 3\hat{j} + 5\hat{k}$

$\vec{B} = 7\hat{i} + 2\hat{j} - x\hat{k}$

Calculate the dot product:

$\vec{A} \cdot \vec{B} = (1)(7) + (-3)(2) + (5)(-x)$

$0 = 7 - 6 - 5x$

$0 = 1 - 5x$

$5x = 1$

$x = \frac{1}{5}$

Thus, the value of $x$ is $\frac{1}{5}$.