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Home » \binom62 \times \binom92 - American Beagle Club

\binom62 \times \binom92 - American Beagle Club

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Question: Find the vector \(\mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\) such that \(\mathbf{v} \times \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 5 \\ 7 \\ -4 \end{pmatrix}\).

📌 Solution: The cross product \(\mathbf{v} \times \mathbf{a}\) is computed as:

📌 \begin{pmatrix} b \cdot 3 - c \cdot (-1) \\ c \cdot 2 - a \cdot 3 \\ a \cdot (-1) - b \cdot 2 \end{pmatrix} = \begin{pmatrix} 3b + c \\ 2c - 3a \\ -a - 2b \end{pmatrix}.

📌 Set this equal to \(\begin{pmatrix} 5 \\ 7 \\ -4 \end{pmatrix}\) and solve the system:

📌 \(3b + c = 5\),

📌 \(2c - 3a = 7\),

📌 \(-a - 2b = -4\).

📌 From equation 3: \(a = 4 - 2b\). Substitute into equation 2: \(2c - 3(4 - 2b) = 7 \implies 2c - 12 + 6b = 7 \implies 2c + 6b = 19\). From equation 1: \(c = 5 - 3b\). Substitute into \(2c + 6b = 19\): \(2(5 - 3b) + 6b = 10 - 6b + 6b = 10 = 19\), which is a contradiction. Thus, no solution exists.

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