# Homework #1

Calculate problems #1 and #2 manually and use http://www.continuummechanics.org/interactivecalcs.html to double-check.

1. Calculate the length of each vector and find the angle between them: $${\bf a} = (12, 3, 4)$$ and $${\bf b} = (16, 48, 12)$$.

2. Find the area of a triangle whose edges are the two vectors in #1 above. Ignore units.

Don't do the remaining ones manually. It's too tedious. Just use the above webpage (or Matlab, etc) and write out the results.

3. Given $$\quad {\bf A} = \left[ \matrix { 2 & 5 & 1 \\ 4 & 8 & 2 \\ 6 & 2 & 4 } \right] \quad$$ and $$\quad {\bf B} = \left[ \matrix { 3 & 4 & 2 \\ 1 & 7 & 5 \\ 3 & 2 & 4 } \right] \quad$$ Demonstrate that $${\bf A} \cdot {\bf B} \ne {\bf B} \cdot {\bf A}$$, but $$( {\bf A} \cdot {\bf B} )^T = {\bf B}^T \cdot {\bf A}^T$$.

4. Calculate the double dot product of the two matrices, $${\bf A} : {\bf B}$$.

5. Calculate the inverse of $${\bf A}$$ in #3 and confirm that $${\bf A} \cdot {\bf A}^{-1} \; = \; {\bf A}^{-1} \cdot {\bf A} \; = \; {\bf I}$$.