\|\mathbf{r} \times \mathbf{s}\| = \sqrt{9^2 + (-2)^2 + 5^2} = \sqrt{81 + 4 + 25} = \sqrt{110}
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- = \mathbf{i}(8 + 1) - \mathbf{j}(4 - 2) + \mathbf{k}(1 + 4)
- = 9\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} = \begin{pmatrix} 9 \\ -2 \\ 5 \end{pmatrix}
- Now compute its magnitude:
- Thus, the area is:
- \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.
