Cross product in uneven matrices

Question

I don't need help with dot product, only the cross product section. Even a hint as to where to start would be great.

Answer 1

You just need to remember these rules: For the standard basis vectors $i$, $j$, and $k$, and for any vectors $\vec v$, $\vec u$, and $\vec w$ and any real number $c$, we have $$\begin{array}\ 1)\ i\times j = k & j\times k = i & k\times i = j\end{array} \\ \begin{array}\ 2)\ \vec v\times \vec u = -\vec u\times \vec v\end{array} \\ \begin{array}\ 3)\ \vec v \times \vec v = 0\end{array} \\ \begin{array}\ 4)\ \vec v\times (\vec u+\vec w) = \vec v\times \vec u + \vec v\times \vec w\end{array} \\ \begin{array}\ 5)\ (c\vec v)\times \vec w = \vec v\times (c\vec w) = c(\vec v\times \vec w) \end{array}$$

Then write out your two vectors in terms of $i$, $j$, and $k$: $A = 2i + j + 3k$ and $B = -i + 2j + 3k$. Then just use your rules:

$$A \times B = (2i + j + 3k)\times (-i + 2j + 3k)$$ $$= -2i\times i +4i\times j +6i\times k -j\times i +2j\times j +3j\times k -3k\times i +6k\times j + 9 k\times k$$ $$= (4+1)i\times j+(6+3)i \times k + (3-6)j\times k$$ $$= 5k -9j-3i$$

Thus the solution is $\begin{bmatrix} -3 \\ -9 \\ 5\end{bmatrix}$.