Two vectors are orthogonal if their dot product is zero:
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- Thus, the solutions are:
- \boxed{30^\circ, 90^\circ, 150^\circ}
- Question: Find the value of $x$ so that the vectors $\begin{pmatrix} 2 \\ x \\ 3 \end{pmatrix}$ and $\begin{pmatrix} x \\ -1 \\ 4 \end{pmatrix}$ are orthogonal.
- \begin{pmatrix} 2 \\ x \\ 3 \end{pmatrix} \cdot \begin{pmatrix} x \\ -1 \\ 4 \end{pmatrix} = 2x + x(-1) + 3(4) = 2x - x + 12 = x + 12 = 0
- Solving:
- x + 12 = 0 \Rightarrow x = \boxed{-12}
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