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Home » Question: Expand the product \( (2x - 3y)^2 \). - IX Labs

Question: Expand the product \( (2x - 3y)^2 \). - IX Labs

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Solution: Two vectors are orthogonal if their dot product is zero. Compute the dot product: \(3x + (2)(-1) + (-1)(4) = 3x - 2 - 4 = 3x - 6\). Set this equal to zero: \(3x - 6 = 0 \implies x = 2\).

📌 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:

📌 \(3b + c = 5\),

📌 \(2c - 3a = 7\),

📌 \(-a - 2b = -4\).

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