A civil engineer analyzing directional stability in solar panel arrays needs to find a vector $\mathbf{v}$ such that $\mathbf{v} imes egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}$, where the cross product is defined in 3D. Find a suitable $\mathbf{v}$.

February 22, 2026

$A civil engineer analyzing directional stability in solar panel arrays needs to find a vector $\mathbf{v}$ such that $\mathbf{v} imes egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}$, where the cross product is defined in 3D. Find a suitable $\mathbf{v}$.$

Scaling by 2 multiplies the entire transformation matrix by 2:
\mathbf{M} = 2 egin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} = egin{pmatrix} 0 & -2 \ 2 & 0 \end{pmatrix}
Thus, the required matrix is $oxed{egin{pmatrix} 0 & -2 \ 2 & 0 \end{pmatrix}}$.
Let $\mathbf{v} = egin{pmatrix} x \ y \ z \end{pmatrix}$, $\mathbf{a} = egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}$, and $\mathbf{v} imes \mathbf{a} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}$.
\mathbf{v} imes \mathbf{a} = egin{vmatrix}
\end{vmatrix} = \mathbf{i}(y \cdot 3 - z \cdot 2) - \mathbf{j}(x \cdot 3 - z \cdot 1) + \mathbf{k}(x \cdot 2 - y \cdot 1)

A civil engineer analyzing directional stability in solar panel arrays needs to find a vector \(\mathbf{v}\) such that \(\mathbf{v} imes egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\), where the cross product is defined in 3D. Find a suitable \(\mathbf{v}\).

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