given v= 5i and w= 2i+3j find the angle between v and w

\(\mbox{The reliable way to do this is to use the formula }\\ \theta = \arccos\left(\dfrac{v \cdot w}{\|v\| \|w\|}\right)\)

\(v=(5,0), \|v\|=5 \\ w=(2,3), \|w\|=\sqrt{2^2+3^2} = \sqrt{13}\\ v\cdot w = 5\cdot 2 + 0\cdot 3 = 10\\ \arccos\left(\dfrac {10}{5 \cdot \sqrt{13}}\right) = 56.31^\circ\)

Thanks Ron. It does help.

Thanks, Rom....here's another way that would work in this case....

5i   =  (5, 0)     this  is a vector that lies on the x axis

2i + 3j  = (2,3)  this is the terminal point of the second vector  

So....we're actually just looking for the angle between (2,3) and the x axis.....this is given  by :

arctan (3/2)   =  56.31°

[ Rom's general formula is usually better  ]