Answer :

Correct answer is : 8.32

The magnitude of the component of $\overrightarrow{a}$ orthogonal to |$\overrightarrow{b}$| = | |$\overrightarrow{a}$| sinθ|

From the dot product of vector :

$cos(\theta )=\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$

$\overrightarrow{\mathrm{a}}=5\mathrm{i}+7\mathrm{j}+2\mathrm{k}$, $\overrightarrow{\mathrm{b}}=3\mathrm{i}-\mathrm{j}+6\mathrm{k}$

a.b = 15 - 7 + 12 = 20

|a| = $\surd 5^2+7^2+2^2=\surd 78$

$|b|=\surd 3^2+(-1{)}^2+6^2=\surd 46$

$cos(\theta )=\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}=\frac{20}{\surd 78\surd 46}$ = 0.334

θ = 70.48

|$\overrightarrow{b}$| = | |$\overrightarrow{a}$| sinθ| = |$\sqrt{78}$. sin(70.49) |

|$\overrightarrow{b}$| = 8.32