Answer
a) $\sqrt {20}$
b) $\sqrt {50}$
c) $\sqrt {50}$
d) $1$
e) $1$
f) $1$
Work Step by Step
$u = \lt 2 , -4\gt$
$v = \lt 5, 5 \gt$
a) $|| u ||$
$= \sqrt {(2)^{2} + (-4)^{2}}$
$= \sqrt {4+16}$
$= \sqrt {20}$
b) $||v||$
$= \sqrt {5^{2}+5^{2}}$
$= \sqrt {25+25}$
$= \sqrt {50}$
c) $||u + v||$
$= ||\lt2,-4\gt + \lt5,5\gt||$
$= ||\lt7,1\gt||$
$= \sqrt {7^{2} + 1^{2}}$
$= \sqrt {49 + 1}$
$= \sqrt {50}$
d) $||\frac{u}{||u||}||$
$= ||\frac{\lt2,-4\gt}{\sqrt {20}}||$
$= ||\lt \frac{2}{\sqrt {20}}, \frac{-4}{\sqrt {20}}\gt||$
$= \sqrt {(\frac{2}{\sqrt {20}})^{2} + (\frac{-4}{\sqrt {20}})^{2}}$
$= \sqrt {0.2 + 0.8}$
$= \sqrt 1$
$ = 1$
e) $||\frac{v}{||v||}||$
$= ||\frac{\lt 5,5\gt}{\sqrt {50}}||$
$= ||\lt \frac{5}{\sqrt {50}}, \frac{5}{\sqrt {50}}\gt||$
$= \sqrt {(\frac{5}{\sqrt {50}})^{2} + (\frac{5}{\sqrt {50}})^{2}}$
$= \sqrt {0.5 + 0.5}$
$= \sqrt 1$
$= 1$
f) $||\frac{u+v}{||u+v||}||$
$= || \frac{\lt 7 ,1\gt}{\sqrt {50}}||$
$= ||\lt \frac{7}{\sqrt {50}}, \frac{1}{\sqrt {50}} \gt||$
$= \sqrt {(\frac{7}{\sqrt {50}})^{2} + (\frac{1}{\sqrt {50}})^{2}}$
$= \sqrt {0.98 + 0.02}$
$= \sqrt 1$
$= 1$
