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

Find the volume of the parallelepiped with edges \((2,0,0),(0,3,0),(0,0,4)\).

Compute \([\mathbf a,\mathbf b,\mathbf c]\) for \(\mathbf a=(1,2,3)\), \(\mathbf b=(0,1,0)\), \(\mathbf c=(2,0,1)\).

Find the volume determined by \((1,1,0),(1,0,1),(0,1,1)\).

Use a scalar triple product to test whether \((1,0,0),(0,1,0),(1,1,0)\) are coplanar.

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

Find the oriented volume sign for \(\mathbf i,\mathbf j,\mathbf k\).

Find \(k\) if \((1,2,3),(2,1,0),(1,0,k)\) are coplanar.

Find the volume of the tetrahedron with edge vectors \((1,0,0),(0,2,0),(0,0,3)\) from one vertex.

Compute the determinant form of the scalar triple product for rows \((1,2,0),(3,1,1),(0,2,4)\).

Show that swapping two vectors changes the sign for \((\mathbf i,\mathbf j,\mathbf k)\).

Find the volume spanned by \((2,0,1),(1,3,0),(0,1,4)\).

Test whether \((1,2,3),(2,4,6),(0,1,1)\) are linearly independent.

Find the scalar triple product \(\mathbf a\cdot(\mathbf b\times\mathbf c)\) for \(\mathbf a=(0,2,1)\), \(\mathbf b=(1,-1,0)\), \(\mathbf c=(3,0,2)\).

Find the area-times-height interpretation for triple product value \(-18\).

For \(\mathbf a=(1,1,1)\), \(\mathbf b=(1,0,2)\), \(\mathbf c=(0,1,t)\), find \(t\) for volume \(3\).

Find the oriented volume of the parallelepiped with edges \((1,2,3),(4,5,6),(7,8,9)\), and explain what the result says about the three edges.

A box has edge vectors \((2,0,0),(1,3,0),(0,1,4)\). Find its volume.

For \(\mathbf a=(a_1,a_2,a_3)\) and \(\mathbf b=(b_1,b_2,b_3)\), show by component calculation that \(\mathbf a\cdot(\mathbf b\times\mathbf a)=0\).

Find \(x\) if the parallelepiped volume from \((x,0,0),(0,2,0),(0,0,5)\) is \(30\).