Answer
a) $\frac{\sqrt 5}{2}$
b) $\sqrt {13}$
c) $ \frac{\sqrt {85}}{2}$
d) $1$
e) $1$
f) $1$
Work Step by Step
$u = \lt 1 , \frac{1}{2}\gt$
$v = \lt 2, 3 \gt$
a) $|| u ||$
$= \sqrt {1^{2}+(0.5)^{2}}$
$= \sqrt {1+0.25}$
$= \sqrt {1.25}$
$= \sqrt {\frac{5}{4}}$
$= \frac{\sqrt 5}{\sqrt 4}$
$= \frac{\sqrt 5}{2}$
b) $||v||$
$= \sqrt {2^{2}+(3)^{2}}$
$= \sqrt {4+9}$
$= \sqrt {13}$
c) $||u + v||$
$= ||\lt 1, \frac{1}{2}\gt + \lt2,3\gt||$
$= ||\lt3, \frac{7}{2}\gt||$
$= \sqrt {(3)^{2}+(\frac{7}{2})^{2}}$
$= \sqrt {9+12.25}$
$= \sqrt {\frac{85}{4}}$
$= \frac{\sqrt {85}}{\sqrt 4}$
$= \frac{\sqrt {85}}{2}$
d) $||\frac{u}{||u||}||$
$= ||\frac{\lt1, \frac{1}{2}\gt}{\frac{\sqrt 5}{2}}||$
$= ||\lt \frac{2}{\sqrt 5}, \frac{1}{\sqrt 5}\gt||$
$= \sqrt {(\frac{2}{\sqrt 5})^{2}+(\frac{1}{\sqrt 5})^{2}}$
$= \sqrt {0.8+0.2}$
$= 1$
e) $||\frac{v}{||v||}||$
$= ||\frac{\lt 2,3\gt}{\sqrt {13}}||$
$= ||\lt \frac{2}{\sqrt {13}}, \frac{3}{\sqrt {13}}\gt||$
$= \sqrt {(\frac{2}{\sqrt {13}})^{2}+(\frac{3}{\sqrt {13}})^{2}}$
$= \sqrt {(0.308...)+(0.692...)}$
$= \sqrt 1$
$= 1$
f) $||\frac{u+v}{||u+v||}||$
$= ||\frac{\lt3, \frac{7}{2}\gt}{\frac{\sqrt {85}}{2}}||$
$= ||\lt \frac{6}{\sqrt {85}}, \frac{7}{\sqrt {85}}\gt||$
$= \sqrt {(\frac{6}{\sqrt {85}})^{2}+(\frac{7}{\sqrt {85}})^{2}}$
$= \sqrt {(0.424...)+(0.576...)}$
$= \sqrt 1$
$= 1$
