Consider two vectors

$\overrightarrow{\mathrm{a}}=5\mathrm{i}+7\mathrm{j}+2\mathrm{k}$

$\overrightarrow{\mathrm{b}}=3\mathrm{i}-\mathrm{j}+6\mathrm{k}$

Magnitude of the component of $\overrightarrow{a}$ orthogonal to $\overrightarrow{b}$ in the plane containing the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______ (round off to 2 decimal places).

The correct answer is 8.32.

To find the magnitude of the component of vector $\overrightarrow{a}$ orthogonal to vector $\overrightarrow{b}$ in the plane containing both vectors, we can use the geometric relationship between the two vectors. Let $\theta$ be the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. The component of $\overrightarrow{a}$ that is perpendicular (orthogonal) to $\overrightarrow{b}$ has a magnitude given by:
$a_{\perp }=\left|\overrightarrow{a}\right|\sin \left(\theta \right)$

First, we write down the components of the given vectors:
$\overrightarrow{a}=5\overrightarrow{i}+7\overrightarrow{j}+2\overrightarrow{k}$
$\overrightarrow{b}=3\overrightarrow{i}-\overrightarrow{j}+6\overrightarrow{k}$

Step 1: Calculate the dot product $\overrightarrow{a}·\overrightarrow{b}$:
$\overrightarrow{a}·\overrightarrow{b}=\left(5\right)\left(3\right)+\left(7\right)(-1)+\left(2\right)\left(6\right)=15-7+12=20$

Step 2: Calculate the magnitudes of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$:
$\left|\overrightarrow{a}\right|=\sqrt{5^2+7^2+2^2}=\sqrt{25+49+4}=\sqrt{78}\approx 8.8318$
$\left|\overrightarrow{b}\right|=\sqrt{3^2+{(-1)}^2+6^2}=\sqrt{9+1+36}=\sqrt{46}\approx 6.7823$

Step 3: Determine the cosine of the angle $\theta$ between the two vectors:
$\cos \left(\theta \right)=\frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}=\frac{20}{\sqrt{78}\sqrt{46}}=\frac{20}{\sqrt{3588}}\approx \frac{20}{59.90}\approx 0.334$

Step 4: Find the angle $\theta$:
$\theta ={\cos }^{-1}\left(0.334\right)\approx {70.48}^{\circ }$

Step 5: Compute the magnitude of the orthogonal component:
$a_{\perp }=\left|\overrightarrow{a}\right|\sin \left(\theta \right)=\sqrt{78}·\sin \left({70.49}^{\circ }\right)\approx 8.8318·0.9426\approx 8.32$