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For each of the following pairs of vectors,
determine a vector orthogonal to both of them.
\[\begin{align*} \bm{u} &= \langle 1,2,3 \rangle \\ \bm{v} &= \langle 3,4,5 \rangle \end{align*}\]\[\begin{align*} \bm{u} &= \langle 1,-2,-3 \rangle \\ \bm{v} &= \langle 3,-4,5 \rangle \end{align*}\]\[\begin{align*} \bm{u} &= \langle 8,-8,1 \rangle \\ \bm{v} &= \langle 0,8,-1 \rangle \end{align*}\]\[\begin{align*} \bm{u} &= \langle 4,5,-2 \rangle \\ \bm{v} &= \langle 3,-5,-1 \rangle \end{align*}\]\[\begin{align*} \bm{u} &= \langle 4,2,1 \rangle \\ \bm{v} &= \langle -1,3,-7 \rangle \end{align*}\]
- For any three non-colinear points in space, there is a unique plane passing through those points, and a unique direction normal (orthogonal) to that plane. What is an example of a vector normal to the plane that contains \((1,2,3)\) and \((1,-3,-5)\) and \((6,1,1)?\)
- For any three non-colinear points in space, there is a unique triangle having those points as vertices. What is the area of the triangle with vertices \((1,2,3)\) and \((-3,3,-2)\) and \((6,1,1)?\) What is the perimeter of this triangle?
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Compute the scalar triple product of each of the following triples of vectors.
\[\begin{align*} \bm{u} &= \langle 1,2,3 \rangle \\ \bm{v} &= \langle 2,3,4 \rangle \\ \bm{w} &= \langle 3,4,5 \rangle \end{align*}\]\[\begin{align*} \bm{u} &= \langle 1,-2,-3 \rangle \\ \bm{v} &= \langle 2,-3,4 \rangle \\ \bm{w} &= \langle 3,-4,5 \rangle \end{align*}\]\[\begin{align*} \bm{u} &= \langle 5,4,-3 \rangle \\ \bm{v} &= \langle 8,8,1 \rangle \\ \bm{w} &= \langle -2,0,7 \rangle \end{align*}\]
- For any four non-coplanar points in space, there is a unique tetrahedron having those points as vertices. What is the volume of the tetrahedron with vertices \((1,2,3)\) and \((-3,3,-2)\) and \((-1,-1,1)\) and \((6,1,1)?\) What is the surface area of this tetrahedron?