Cross Product of Vectors

Cross Product - Properties


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

Given two vectors such that a=3,b=4,\lvert\vec{a}\rvert=3,\lvert\vec{b}\rvert=4, and ab=8,\vec{a}\cdot\vec{b}=8, what is the magnitude of b×a?\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 τ=r×F,\vec{\tau}=\vec{r}\times\vec{F}, where r\vec{r} is the displacement vector (pointing from the axis to the point at which the force is applied) and F\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 a=(1,1,2),b=(2,1,3),\vec{a}=(1,1,2), \vec{b}=(2,1,3), and c=(3,4,1).\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),O=(0,0,0), A=(4,0,1), B=(-2,-4,3), and C=(3,1,1)?C=(3,-1,-1)?


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