Finding $\vec{v}\times\hat{i},\vec{v}\times\hat{j},\vec{v}\times\hat{k}$

Question

How do I find the following?

\begin{align}\vec{v}\times\hat{i},\\ \vec{v}\times\hat{j},\tag{1} \\ \vec{v}\times\hat{k},\end{align}

given only that \begin{align} \vec{v} = \begin{bmatrix} 9 \\ 3 \\ 2 \end{bmatrix},\tag{2} \end{align} and since I'm not crossing this vector with another vector it seems odd...

Thanks,

Answer 1

$\hat i=(1,0,0),\hat j=(0,1,0),\hat k=(0,0,1)$

Answer 2

We generally take \begin{align*} \hat i &= \begin{bmatrix} 1\\0\\0\end{bmatrix} & \hat j &= \begin{bmatrix} 0\\1\\0\end{bmatrix} & \hat k &= \begin{bmatrix} 0\\0\\1\end{bmatrix} \end{align*} Furthermore, for \begin{align*} \vec x &= \begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix} & \vec y &= \begin{bmatrix} y_1\\y_2\\y_3\end{bmatrix} \end{align*} we define $\vec x\times\vec y$ by the formula $$ \vec x\times\vec y= \begin{bmatrix} \begin{vmatrix}x_2 & x_3\\ y_2 & y_3\end{vmatrix} \\ -\begin{vmatrix}x_1 & x_3\\ y_1 & y_3\end{vmatrix} \\ \begin{vmatrix}x_1 & x_2\\ y_1 & y_2\end{vmatrix} \\ \end{bmatrix} $$ where $$ \begin{vmatrix} a&b\\c&d \end{vmatrix}=ad-bc $$ Can you put all of this together to obtain your desired equations?