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📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 However, the condition $\mathbf{v} \cdot \mathbf{a} = 0$ implies $v_1 + 2v_2 + 3v_3 = 0$. Substituting $v_2 = 6 + 2v_1$ and $v_3 = 11 + 3v_1$:

📌 v_1 + 2(6 + 2v_1) + 3(11 + 3v_1) = 0 \implies v_1 + 12 + 4v_1 + 33 + 9v_1 = 0 \implies 14v_1 + 45 = 0 \implies v_1 = -\frac{45}{14}.

📌 Then $v_2 = 6 + 2(-\frac{45}{14}) = \frac{84 - 90}{14} = -\frac{6}{14} = -\frac{3}{7}$, and $v_3 = 11 + 3(-\frac{45}{14}) = \frac{154 - 135}{14} = \frac{19}{14}$.

📌 Thus, $\mathbf{v} = \left\langle -\frac{45}{14}, -\frac{3}{7}, \frac{19}{14} \right\rangle$.

📌 Verification: Cross product and dot product conditions are satisfied.

📌 \boxed{\left\langle -\dfrac{45}{14}, -\dfrac{3}{7}, \dfrac{19}{14} \right\rangle}

📌 Question: In spherical coordinates, describe the surface defined by $\rho = 2\sin\phi \cos\theta$.

📌 Solution: Convert to Cartesian coordinates using $\rho = \sqrt{x^2 + y^2 + z^2}$, $x = \rho \sin\phi \cos\theta$, $y = \rho \sin\phi \sin\theta$, $z = \rho \cos\phi$. Multiply both sides by $\rho$:

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