Let $v = (3,0,0)$ and $w=(0,1,-1).$ Determine $u = v \times w$ using the geometric properties of the cross product rather than the formula.
What are the possible angles $x$ between two unit vectors $e$ and $f$ if $||e \times f||=1/2$?
For the first question, the way I approached it was by using the cross product before I used the geometric properties of the cross product. So, after doing little bit of algebra for $u = v \times w,$ I computed $= 0i+3j+3k=(0,3,3)$ but something looks wrong. For the second question, I suspect that the answer is $\frac{pi}{2}$ but don't know how to show it. Much appreciated.
I highly recommend reading the wiki on the geometric interpretation of the cross product for part one. For part two, you will realize your answer is incorrect by using the same geometric interpretation: if two unit vectors have angle $\frac{\pi}{2}$, then the magnitude of their cross product must equal one. You can also directly compute the result using $||a\times b|| = ||a||||b||\,sin{\theta}$