The Cross Product

  1. For each of the following pairs of vectors, determine a vector orthogonal to both of them.
    u=⟨1,2,3⟩v=⟨3,4,5⟩\begin{align*} \bm{u} &= \langle 1,2,3 \rangle \\ \bm{v} &= \langle 3,4,5 \rangle \end{align*}
    u=⟨1,−2,−3⟩v=⟨3,−4,5⟩\begin{align*} \bm{u} &= \langle 1,-2,-3 \rangle \\ \bm{v} &= \langle 3,-4,5 \rangle \end{align*}
    u=⟨8,−8,1⟩v=⟨0,8,−1⟩\begin{align*} \bm{u} &= \langle 8,-8,1 \rangle \\ \bm{v} &= \langle 0,8,-1 \rangle \end{align*}
    u=⟨4,5,−2⟩v=⟨3,−5,−1⟩\begin{align*} \bm{u} &= \langle 4,5,-2 \rangle \\ \bm{v} &= \langle 3,-5,-1 \rangle \end{align*}
    u=⟨4,2,1⟩v=⟨−1,3,−7⟩\begin{align*} \bm{u} &= \langle 4,2,1 \rangle \\ \bm{v} &= \langle -1,3,-7 \rangle \end{align*}
  2. 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)(1,2,3) and (1,−3,−5)(1,-3,-5) and (6,1,1)?(6,1,1)?
  3. 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)(1,2,3) and (−3,3,−2)(-3,3,-2) and (6,1,1)?(6,1,1)? What is the perimeter of this triangle?
  4. Compute the scalar triple product of each of the following triples of vectors.
    u=⟨1,2,3⟩v=⟨2,3,4⟩w=⟨3,4,5⟩\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*}
    u=⟨1,−2,−3⟩v=⟨2,−3,4⟩w=⟨3,−4,5⟩\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*}
    u=⟨5,4,−3⟩v=⟨8,8,1⟩w=⟨−2,0,7⟩\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*}
  5. 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)(1,2,3) and (−3,3,−2)(-3,3,-2) and (−1,−1,1)(-1,-1,1) and (6,1,1)?(6,1,1)? What is the surface area of this tetrahedron?