Answer
a) $1$
b) $\sqrt {18}$
c) $\sqrt {13}$
d) $1$
e) $1$
f) $1$
Work Step by Step
$u = \lt 0 , 1\gt$
$v = \lt 3, -3 \gt$
a) $|| u ||$
$= \sqrt {0^{2} + 1^{2}}$
$= \sqrt {0 + 1}$
$= 1$
b) $||v||$
$= \sqrt {3^{2} + (-3)^{2}}$
$= \sqrt {9 + 9}$
$= \sqrt {18}$
c) $||u + v||$
$=|| \lt 0 ,1\gt + \lt 3, -3\gt||$
$= ||\lt 3, -2\gt||$
$= \sqrt {3^{2} + (-2)^{2}}$
$= \sqrt {9 + 4}$
$= \sqrt {13}$
d) $||\frac{u}{||u||}||$
$= ||\frac{\lt 0,1\gt}{1}||$
$= ||\lt 0 ,1\gt||$
$= \sqrt {0^{2} + (1)^{2}}$
$= \sqrt {0+1}$
$= 1$
e) $||\frac{v}{||v||}||$
$= || \frac{\lt3,-3\gt}{\sqrt 18}||$
$= ||\lt \frac{3}{\sqrt {18}}, \frac{-3}{\sqrt {18}}\gt||$
$= \sqrt {(\frac{3}{\sqrt {18}})^{2} + (\frac{-3}{\sqrt {18}})^{2}}$
$= \sqrt {0.5 + 0.5}$
$= \sqrt 1$
$= 1$
f) $||\frac{u+v}{||u+v||}||$
$= ||\frac{\lt3,-2\gt}{\sqrt {13}}||$
$= ||\lt\frac{3}{\sqrt {13}} , \frac{-2}{\sqrt {13}}\gt||$
$= \sqrt {(\frac{3}{\sqrt {13}})^{2}+(\frac{-2}{\sqrt {13}})^{2}}$
$= \sqrt {(0.692...) + (0.308...)}$
$= \sqrt 1$
$= 1$
