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Home » Relive Paradise with Take Me Back to Eden Lyrics – Could These Words Change Your Mind? - Crosslake

Relive Paradise with Take Me Back to Eden Lyrics – Could These Words Change Your Mind? - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: The given points lie on the plane \(z = 3\). The vector from \((1, 2, 3)\) to \((1, 5, 3)\) is \((0, 3, 0)\), and the vector to \((4, 2, 3)\) is \((3, 0, 0)\). The fourth vertex on the same face is obtained by adding these vectors: \((1 + 3, 2 + 3, 3) = (4, 5, 3)\).

📌 \boxed{(4, 5, 3)}

📌 Question: Find the value of \(x\) such that the vectors \(\begin{pmatrix} x \\ 2 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}\) are orthogonal.

📌 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:

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