🎥 Video solution / Text Solution of this question is given below:
Given:
$\vec{a} = \hat{i} + \hat{j} + \hat{k}$
$\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$
$\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$
Compute:
$2\vec{a} - \vec{b} + 3\vec{c}$
First:
$2\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$
$-\vec{b} = -2\hat{i} + \hat{j} - 3\hat{k}$
$3\vec{c} = 3\hat{i} - 6\hat{j} + 3\hat{k}$
Add components:
$i$–component: $2 - 2 + 3 = 3$
$j$–component: $2 + 1 - 6 = -3$
$k$–component: $2 - 3 + 3 = 2$
So the vector:
$\vec{v} = 3\hat{i} - 3\hat{j} + 2\hat{k}$
Its magnitude:
$|\vec{v}| = \sqrt{3^2 + (-3)^2 + 2^2} = \sqrt{9 + 9 + 4} = \sqrt{22}$
We need a vector parallel to $\vec{v}$ with magnitude $\sqrt{22}$.
Since $|\vec{v}| = \sqrt{22}$, the required vector is simply:
$\boxed{3\hat{i} - 3\hat{j} + 2\hat{k}}$