Score: $9.5 / 10 \quad 9 / 10$ answered
Question 7
Find two unit vectors orthogonal to both $\vec{u}=\langle 3,4,-2\rangle$ and $\vec{v}=\langle 0,5,3\rangle$. 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 \(\vec{u} = \langle 3,4,-2 \rangle\) and \(\vec{v} = \langle 0,5,3 \rangle\) by taking their cross product. The cross product of \(\vec{u}\) and \(\vec{v}\) is \(\langle 22, -9, 15 \rangle\).

Step 2 ：Next, we normalize this vector to get a unit vector. The normalized vector is \(\langle 0.78272487, -0.32020563, 0.53367605 \rangle\).

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

Step 4 ：Finally, we normalize this vector to get the second unit vector. The normalized vector is \(\langle -0.27748323, 0.58800018, 0.75977552 \rangle\).

Step 5 ：The two unit vectors orthogonal to both \(\vec{u} = \langle 3,4,-2 \rangle\) and \(\vec{v} = \langle 0,5,3 \rangle\) are \(\boxed{\langle 0.78272487, -0.32020563, 0.53367605 \rangle}\) and \(\boxed{\langle -0.27748323, 0.58800018, 0.75977552 \rangle}\).