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Question 7
Find two unit vectors orthogonal to both $\overrightarrow{u}=⟨3,2,-1⟩$ and $\overrightarrow{v}=⟨0,1,5⟩$. Give exact values (no decimals). Separate the vectors with a comma.
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Step 1 ：Given vectors $\overrightarrow{u}=⟨3,2,-1⟩$ and $\overrightarrow{v}=⟨0,1,5⟩$

Step 2 ：Find the cross product of $\overrightarrow{u}$ and $\overrightarrow{v}$ to get a vector $\overrightarrow{w}$ orthogonal to both. The cross product is given by the determinant of the following matrix: $$\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 2& -1\\ 0& 1& 5\end{array}\right]$$

Step 3 ：The cross product is then $\overrightarrow{w}=⟨2\ast 5-(-1)\ast 1,-1\ast 5-3\ast 0,3\ast 1-2\ast 0⟩=⟨11,-5,3⟩$

Step 4 ：Normalize this vector by dividing each component by the magnitude of the vector. The magnitude of $\overrightarrow{w}$ is $\sqrt{{11}^2+(-5{)}^2+3^2}=\sqrt{135}$

Step 5 ：The unit vector orthogonal to both $\overrightarrow{u}$ and $\overrightarrow{v}$ is $\overrightarrow{w_1}=⟨\frac{11}{\sqrt{135}},\frac{-5}{\sqrt{135}},\frac{3}{\sqrt{135}}⟩$

Step 6 ：The second unit vector orthogonal to both $\overrightarrow{u}$ and $\overrightarrow{v}$ is simply the negative of the first, $\overrightarrow{w_2}=⟨\frac{-11}{\sqrt{135}},\frac{5}{\sqrt{135}},\frac{-3}{\sqrt{135}}⟩$

Step 7 ：$\boxed{{\displaystyle \text{Final Answer: The two unit vectors orthogonal to both }\overrightarrow{u}=⟨3,2,-1⟩\text{ and }\overrightarrow{v}=⟨0,1,5⟩\text{ are }\overrightarrow{w_1}=⟨\frac{11}{\sqrt{135}},\frac{-5}{\sqrt{135}},\frac{3}{\sqrt{135}}⟩\text{ and }\overrightarrow{w_2}=⟨\frac{-11}{\sqrt{135}},\frac{5}{\sqrt{135}},\frac{-3}{\sqrt{135}}⟩}}$