# Matrix transformations and effects on the unit square

 Homework Sci Advisor HW Helper Thanks P: 12,463 Matrix transformations and effects on the unit square The advantage of the vector form is that it gives you the direction of the cross product as well. Consider if ##\vec{v} = (v\cos\phi,v\sin\phi,0)^t## and ##\vec{u}=(u\cos\theta, u\sin\theta, 0)^t## Where ##\theta## is the angle ##\vec{u}## makes to the x axis and ##\phi## is the angle ##\vec v## makes to the x axis. Then $$\vec{u}\times\vec{v} = \left| \begin{array}{ccc} \hat{\imath} & \hat{\jmath} & \hat{k}\\ u\cos\theta & u\sin\theta & 0\\ v\cos\phi & v\cos\phi & 0\end{array}\right| = uv\sin(\phi-\theta)\hat{k}$$ ... which result required trig identities. Notice that ## \theta -\phi ## is the angle between the vectors - so the equation you are used to is just for the magnitude. It gets trickier when you go to more than three dimensions.