\mathbf{f} \cdot \mathbf{g} = 5(-1) + k(3) + 2(6) = -5 + 3k + 12 = 3k + 7 = 0
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- \boxed{\sqrt{110}}
- An angel investor assesses two biotech trends represented by vectors $\mathbf{f} = \begin{pmatrix} 5 \\ k \\ 2 \end{pmatrix}$ and $\mathbf{g} = \begin{pmatrix} -1 \\ 3 \\ 6 \end{pmatrix}$, seeking $k$ such that $\mathbf{f}$ and $\mathbf{g}$ are orthogonal.
- Vectors are orthogonal if their dot product is zero:
- Solve for $k$:
- 3k = -7 \Rightarrow k = -\frac{7}{3}
- Thus, the required value is:
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