I'm reading a textbook and it's going over finding the dot product of two vectors:
$$u * v = \|u\|*\|v\|*\cos\theta$$
The vectors are:
$$u = (0, 0, 1) \\
v = (0, 2, 2)$$
With lengths:
$$\|u\| = 1 \\
\|v\| = \sqrt{8}$$
However, in the text it jumps a little and I'm not sure how they went from $\sqrt{8}$ to $2\sqrt{2}$. What does this mean? $2*\sqrt{2}$ does indeed equal $\sqrt{8}$, but how did it go from one to the other?
Secondly, in the text it then states $cos(45^*) = 1/\sqrt{2}$. I'm not sure how they got to this point either. Where did the righthand portion of that come from? How did they calculate the angle? I lack the foundations of trig/geometry so maybe I should know this, but I just don't.
This is as it appears in the text, without any additional information:
Best Answer
angle between two vector it is equal
$\cos(\theta)=(u*v)/|u|*|v|$
you know lengths and you know that $u*v=0*0+0*2+1*2=2$
now please insert values
you will have
$\cos(\theta)=2/(1*\sqrt{8})=2/(2*\sqrt{2})=1/\sqrt{2}$
now use this table