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Geometry

Cross Product of 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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