Question Type 1: Finding the cross product of two vectors Exercises

Compute the cross product of the vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ 1 \\ 4 \end{pmatrix}$.

Find the cross product of $\mathbf{p} = (2,-1,3)$ and $\mathbf{q} = (4,0,2)$.

Find the cross product of $\mathbf{u} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

Determine a vector perpendicular to $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$.

Evaluate $(3,4,0)\times(0,0,5)$.

Compute the cross product of $\mathbf{p}=(1,0,2)$ and $\mathbf{q}=(3,1,1)$.

Determine whether $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}$ are parallel by computing $\mathbf{a} \times \mathbf{b}$.

Given $\mathbf{a}=\begin{pmatrix}2\\3\\4\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}-1\\0\\2\end{pmatrix}$, find the magnitude $\|\mathbf{a}\times\mathbf{b}\|$.

Compute the area of the parallelogram spanned by $\mathbf{a}=\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}$ by finding $\|\mathbf{a}\times\mathbf{b}\|$.

Find $(-1, 2, 5) \times (3, 1, -2)$.

If $\mathbf{a}=(a,0,1)$ and $\mathbf{b}=(0,b,2)$, find $\mathbf{a}\times\mathbf{b}$ in terms of $a$ and $b$.

If $\mathbf{u}=(t,2,3)$ and $\mathbf{v}=(1,t,4)$, compute $\mathbf{u}\times\mathbf{v}$ in terms of $t$.