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Home » Express \( w \) in terms of \( l \): - jntua results

Express \( w \) in terms of \( l \): - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 A science communicator is creating a vector visualization for a YouTube video and needs to find a vector \(\mathbf{v} = egin{bmatrix} a \ b \end{bmatrix}\) such that \(\mathbf{v} imes \mathbf{a} = 5\), where \(\mathbf{a} = egin{bmatrix} 3 \ -4 \end{bmatrix}\) and cross product in 2D is defined as \(\mathbf{v} imes \mathbf{a} = a_1b_2 - a_2b_1\).

📌 \mathbf{v} = egin{bmatrix} a \ b \end{bmatrix}, \quad \mathbf{a} = egin{bmatrix} 3 \ -4 \end{bmatrix}

📌 The 2D cross product is:

📌 \mathbf{v} imes \mathbf{a} = a(-4) - b(3) = -4a - 3b

📌 We set this equal to 5:

📌 -4a - 3b = 5

📌 This is a linear equation in two variables. We can express one variable in terms of the other. Solving for \(b\):

📌 -3b = 4a + 5 \quad \Rightarrow \quad b = -rac{4a + 5}{3}

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