Compute \((1,0,0)\times(0,1,0)\).

Compute \((0,1,0)\times(1,0,0)\).

Find \((2,3,4)\times(1,0,2)\).

Find the area of the parallelogram spanned by \((3,0,0)\) and \((0,4,0)\).

Find the area of the triangle with sides \((2,0,0)\) and \((0,5,0)\) from one vertex.

Compute the torque \(\boldsymbol\tau=\mathbf r\times\mathbf F\) for \(\mathbf r=(2,0,0)\) m and \(\mathbf F=(0,3,0)\) N.

Test whether \((1,2,3)\) and \((2,4,6)\) are parallel using a cross product.

Find a vector perpendicular to \((1,1,0)\) and \((0,1,1)\).

Compute \((3,-1,2)\times(2,1,0)\).

Find \(|(1,2,2)\times(2,0,1)|\).

Find the unit normal to the plane spanned by \((1,0,1)\) and \((0,2,0)\).

Compute \((\mathbf i+\mathbf j)\times(\mathbf j+\mathbf k)\).

Find \(k\) if \((1,2,3)\times(k,4,6)=\mathbf0\).

Find the orientation sign of \((1,0,0)\times(0,1,0)\cdot(0,0,1)\).

A triangular plate has vertices \(A=(1,0,2)\), \(B=(2,2,2)\), and \(C=(4,0,-1)\). Find its area and a unit normal vector.

Compute \((2,-1,1)\times(1,3,-2)\).

Show that \((2,-1,1)\times(1,3,-2)\) is perpendicular to \((2,-1,1)\).

Find the sine of the angle between \((1,0,1)\) and \((0,1,1)\), then use it to find the area of the parallelogram they span.

For the triangle with vertices \(A=(0,0,0)\), \(B=(1,2,0)\), and \(C=(0,1,3)\), find a normal vector and decide which side of the triangle the point \(P=(6,-3,1)\) lies on relative to that normal.

A force \((0,5,0)\) N is applied at \((2,0,1)\) m. Find the torque about the origin, its magnitude, and explain whether the torque is perpendicular to the force.