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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!

Concept Quizzes

Challenge Quizzes

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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