Problem with vector multiplication

Question

I have this plane problem and the answers are released for it. I don't understand this specific part: Why does : (i + 4k) x (3j - k) = -12i + j + 3k.

I tried using the cross product method, however, i am stuck and can't proceed on. Please explain the manner in which this is calculated.

Thank you!

Answer 1

Using matrix notation:

$$(i+4k)\times(3j-k)=\begin{vmatrix}i&j&k\\1&0&4\\0&3&\!\!-1\end{vmatrix}=(0-12)i+(0+1)j+(3-0)k=-12i+j+3k$$

Answer 2

First, expand it out:

$$(i + 4k)\times(3j-k) = 3(i\times{j}) + 12(k\times{j}) - (i\times{k}) - 4(k\times{k})$$

But $i\times{j} = k, k\times{j} = -i,i\times{k} = -j, k\times{k} = 0$. These can be deduced from the definition of the standard basis vectors $i,j,k$. Hence,

$$(i + 4k)\times(3j-k) = 3k -12i +j$$

Rearranging,

$$(i + 4k)\times(3j-k) = -12i + j + 3k$$