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# Cross Product of Vectors

The cross product is a fundamental operation on vectors. It acts on vectors in three dimensions and results in another vector in three dimensions which is perpendicular to both of the other vectors!

# Cross Product - Properties

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?

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 $$\vec{b}\times\vec{a}?$$

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?

Find the volume of the parallelepiped spanned by the vectors $$\vec{a}=(1,1,2), \vec{b}=(2,1,3),$$ and $$\vec{c}=(3,4,1).$$

What is the volume of the tetrahedron whose vertices are $$O=(0,0,0), A=(4,0,1), B=(-2,-4,3),$$ and $$C=(3,-1,-1)?$$

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