Question

Let vector $a = (1, sqrt(2) ,1)$ and vector $b = (2,0,2)$

1. Find the angle $\theta$ between the 2 vectors.
2. Find the projection of a in the direction of b

Answer 1

$let\space a=(1,\sqrt2,1)\\b=(2,0,2)\\$

a.

$cos(\theta)=\frac{a.b}{|a|.|b|}\\$

$=\frac{(1,\sqrt2,1).(2,0,2)}{\sqrt{1+2+1}{\sqrt{4+4}}}$

$=\frac{2+2}{\sqrt4 \sqrt8}=\frac{4}{2.2\sqrt2}$

$=\frac{1}{\sqrt2}$

$\theta=cos^{-1}(\frac{1}{\sqrt2})\\\theta=45\degree$

b.

Projection of a in direction of b

$=\frac{\vec{a}.\vec{b}}{|\vec{b}|}\space or\space |\vec{a}|cos\theta$

$\&\space |\vec{b}|=\sqrt8$

$=\frac{4}{\sqrt8}=\frac{4}{2\sqrt2}=\frac{2}{\sqrt2}=\sqrt2$