Consider a vector \(\vec{v}\) of magnitude \(\sqrt{26}.\) If this vector is perpendicular to both \(\vec{a}=(2,3,1)\) and \(\vec{b}=(1,3,5),\) what is the square of its \(x\)-coordinate?

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Given two vectors such that \(\lvert\vec{a}\rvert=3,\lvert\vec{b}\rvert=4,\) and \(\vec{a}\cdot\vec{b}=8,\) what is the magnitude of \(\vec{b}\times\vec{a}?\)

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Torque is vector that measures the tendency of a force to rotate an object about an axis. It is given by the formula
\[\vec{\tau}=\vec{r}\times\vec{F},\]
where \(\vec{r}\) is the displacement vector (pointing from the axis to the point at which the force is applied) and \(\vec{F}\) is the force vector. If a disc rotates counter-clockwise on a record player, as shown in the figure above, what is the direction of the torque?

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