Answer
a) $\sqrt 2$
b) $\sqrt 5$
c) $1$
d) $1$
e) $1$
f) $1$
Work Step by Step
$u = \lt1 , -1\gt$
$v = \lt -1, 2 \gt$
a) $|| u ||$
$= \sqrt {1^{2} + (-1)^{2}}$
$= \sqrt {1 + 1}$
$= \sqrt {2}$
b) $||v||$
$= \sqrt {(-1)^{2} + 2^{2}}$
$= \sqrt {1 + 4}$
$= \sqrt 5$
c) $||u + v||$
$= \lt1 , -1\gt + \lt -1, 2 \gt$
$= \lt 0, 1\gt$
$= \sqrt {0^{2} + 1^{2}}$
$= 1$
d) $||\frac{u}{||u||}||$
$= \frac{ \lt1 , -1\gt$}{\sqrt 2}$
$= \lt \frac{1}{\sqrt {2}}, -\frac{1}{\sqrt 2} \gt$
$= \sqrt {(\frac{1}{\sqrt {2}})^{2} + (-\frac{1}{\sqrt 2})^{2}}$
$= \sqrt {0.5 + 0.5}$
$= \sqrt 1$
$= 1$
e) $||\frac{v}{||v||}||$
$= \frac{\lt -1, 2\gt}{\sqrt 5}$
$= \lt -\frac{1}{\sqrt 5} , \frac{2}{\sqrt 5}\gt$
$= \sqrt {(-\frac{1}{\sqrt 5})^{2}+(\frac{2}{\sqrt 5})^{2}}$
$= \sqrt {0.2 + 0.8}$
$= \sqrt 1$
$= 1$
f) $||\frac{u+v}{||u+v||}||$
$= \frac{\lt 0, 1\gt}{1}$
$= \lt 0, 1\gt$
$= \sqrt {0^{2} + 1^{2}}$
$= \sqrt {1}$
$= 1$
