Anyone come across this before?

[Mathmasteriw]

hi Guys and girls,
Can anyone help me here and explain how you get to the bottom step? I’ve tried few different ways and can’t get it right!
Thank you for your time

 

[HallsofIvy]

Surely, if you have learned the "cross products" in terms of the determinant you learned how to find a determinant, haven't you? You seem to have no problem with "expanding" a determinant by the first row: You should have learned that $\left| \begin{array}{cc} a & b \\ c & d \end{array} \right|= ad- bc$.

 

[pka]

The above is the standard evaluation of a a \(3\times 3\) determent of \(3\times 3\) matrix.
In this case it is expanded along the first row of the the matrix. In this case \(i,~j.~\&~k\) are the three basic vectors of \(\Re^3\).
In this case the result is a vector which is perpendicular to the vectors \(<4,5,-7>~\&~<4,-3,5>\)
Look at this link.