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

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

Step 3 ：The cross product is then \(\vec{w} = \langle 2*5 - (-1)*1, -1*5 - 3*0, 3*1 - 2*0 \rangle = \langle 11, -5, 3 \rangle\)

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

Step 5 ：The unit vector orthogonal to both \(\vec{u}\) and \(\vec{v}\) is \(\vec{w_1} = \langle \frac{11}{\sqrt{135}}, \frac{-5}{\sqrt{135}}, \frac{3}{\sqrt{135}} \rangle\)

Step 6 ：The second unit vector orthogonal to both \(\vec{u}\) and \(\vec{v}\) is simply the negative of the first, \(\vec{w_2} = \langle \frac{-11}{\sqrt{135}}, \frac{5}{\sqrt{135}}, \frac{-3}{\sqrt{135}} \rangle\)

Step 7 ：\(\boxed{\text{Final Answer: The two unit vectors orthogonal to both } \vec{u}=\langle 3,2,-1\rangle \text{ and } \vec{v}=\langle 0,1,5\rangle \text{ are } \vec{w_1} = \langle \frac{11}{\sqrt{135}}, \frac{-5}{\sqrt{135}}, \frac{3}{\sqrt{135}} \rangle \text{ and } \vec{w_2} = \langle \frac{-11}{\sqrt{135}}, \frac{5}{\sqrt{135}}, \frac{-3}{\sqrt{135}} \rangle}\)