The vector in the direction of the vector $\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}}$ that has magnitude 9 is

$3(\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}})$

The correct answer is Option (3) → $3(\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}})$ ##

Let $\mathbf{a} = \hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}}$

Any vector in the direction of a vector $\mathbf{a}$ is given by $\frac{\mathbf{a}}{|\mathbf{a}|}$

$= \frac{\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}}}{\sqrt{1^2 + 2^2 + 2^2}} = \frac{\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}}}{3}$

$∴$ Vector in the direction of $\mathbf{a}$ with magnitude $9 = 9 \cdot \frac{\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}}}{3}$

$= 3(\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}})$