The cross product

Which of the following systems of vectors are right-handed? Which are left-handed?

Show that if $\mathbf{a}\times\mathbf{u}=\mathbf{0}$ for every unit vector $\mathbf{u}$, then $\mathbf{a}=\mathbf{0}$.

Find the area of the parallelogram determined by the vectors $\left[\begin{array}{r}1\\ 2\\ 3\end{array}\right]$, $\left[\begin{array}{r}3\\ -2\\ 1\end{array}\right]$.

Find the area of the parallelogram determined by the vectors $\left[\begin{array}{r}1\\ 0\\ 3\end{array}\right]$, $\left[\begin{array}{r}4\\ -2\\ 1\end{array}\right]$.

Find the area of the parallelogram with vertices $(-2,3,1)$, $(2,1,1)$, $(1,2,-1)$, and $(5,0,-1)$.

Find the area of the triangle determined by the three points, $(1,0,3),(4,1,0)$ and $(-3,1,1)$.

Find the area of the triangle determined by the three points, $(1,2,3),(2,3,4)$ and $(3,4,5)$. Did something interesting happen here? What does it mean geometrically?

Is $\mathbf{u}\times(\mathbf{v}\times\mathbf{w})=(\mathbf{u}\times\mathbf{v})\times\mathbf{w}$? What is the meaning of $\mathbf{u}\times\mathbf{v}\times\mathbf{w}$? Explain. Hint: Try $(\mathbf{i}\times\mathbf{j})\times\mathbf{j}$.

Prove the following formula by direct calculation: $\mathbf{u}\times(\mathbf{v}\times\mathbf{w})~{}=~{}(\mathbf{u}\boldsymbol{\cdot}\mathbf{w})\,\mathbf{v}-(\mathbf{u}\boldsymbol{\cdot}\mathbf{v})\,\mathbf{w}$.

Use the formula from Exercise 47 to prove that the cross product satisfies the so-called Jacobi identity:

Find the volume of the parallelepiped determined by the vectors $\left[\begin{array}{r}1\\ -7\\ -5\end{array}\right]$, $\left[\begin{array}{r}1\\ -2\\ -6\end{array}\right]$, and $\left[\begin{array}{r}3\\ 2\\ 3\end{array}\right]$.

Which of the following systems of vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ are right-handed? Which are left-handed? Which are coplanar?

1. $\mathbf{u}=\left[1,0,1\right]^{T}$, $\mathbf{v}=\left[1,2,0\right]^{T}$, $\mathbf{w}=\left[0,0,1\right]^{T}$.

2. $\mathbf{u}=\left[0,1,1\right]^{T}$, $\mathbf{v}=\left[-1,2,0\right]^{T}$, $\mathbf{w}=\left[1,1,2\right]^{T}$.

3. $\mathbf{u}=\left[1,-1,0\right]^{T}$, $\mathbf{v}=\left[1,0,1\right]^{T}$, $\mathbf{w}=\left[3,1,4\right]^{T}$.

4. $\mathbf{u}=\left[1,0,0\right]^{T}$, $\mathbf{v}=\left[1,2,0\right]^{T}$, $\mathbf{w}=\left[2,0,-1\right]^{T}$.

Suppose $\mathbf{u},\mathbf{v}$, and $\mathbf{w}$ are three vectors whose components are all integers. Can you conclude the volume of the parallelepiped determined from these three vectors will always be an integer?

What does it mean geometrically if the box product of three vectors equals zero?

Use the formula from Exercise 47 to show that

Simplify $\left\|\mathbf{u}\times\mathbf{v}\right\|^{2}+(\mathbf{u}\boldsymbol{\cdot}\mathbf{v})^{2}-\left\|\mathbf{u}\right\|^{2}\left\|\mathbf{v}\right\|^{2}$.