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Question 7
Find two unit vectors orthogonal to both $\overrightarrow{u}=⟨3,4,-2⟩$ and $\overrightarrow{v}=⟨0,5,3⟩$. Give exact values (no decimals). Separate the vectors with a comma.
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Step 1 ：First, we find a vector that is orthogonal to both $\overrightarrow{u}=⟨3,4,-2⟩$ and $\overrightarrow{v}=⟨0,5,3⟩$ by taking their cross product. The cross product of $\overrightarrow{u}$ and $\overrightarrow{v}$ is $⟨22,-9,15⟩$.

Step 2 ：Next, we normalize this vector to get a unit vector. The normalized vector is $⟨0.78272487,-0.32020563,0.53367605⟩$.

Step 3 ：Then, we find a second vector that is orthogonal to both $\overrightarrow{u}$ and $\overrightarrow{v}$ by taking the cross product of the first unit vector we found and either $\overrightarrow{u}$ or $\overrightarrow{v}$. The cross product is $⟨-1.49429294,3.1664779,4.09151639⟩$.

Step 4 ：Finally, we normalize this vector to get the second unit vector. The normalized vector is $⟨-0.27748323,0.58800018,0.75977552⟩$.

Step 5 ：The two unit vectors orthogonal to both $\overrightarrow{u}=⟨3,4,-2⟩$ and $\overrightarrow{v}=⟨0,5,3⟩$ are $\boxed{{\displaystyle ⟨0.78272487,-0.32020563,0.53367605⟩}}$ and $\boxed{{\displaystyle ⟨-0.27748323,0.58800018,0.75977552⟩}}$.