Answer
(a) $-\frac{14\sqrt {97}}{97}$
(b) $-\frac{56}{97}i+\frac{126}{97}j$
(c) $-\frac{56}{97}i+\frac{126}{97}j$, $\frac{153}{97}i+\frac{68}{97}j$
Work Step by Step
(a) Given $\vec u=i+2j=\langle 1, 2 \rangle$ and $\vec v=4i-9j=\langle 4, -9 \rangle$，we have
$comp_v\vec u=\frac{\vec u\cdot\vec v}{|\vec v|}=\frac{1\times4+2\times(-9)}{\sqrt {4^2+(-9)^2}}=-\frac{14}{\sqrt {97}}=-\frac{14\sqrt {97}}{97}$
(b) $proj_v\vec u=(\frac{\vec u\cdot\vec v}{|\vec v|^2})\vec v=-\frac{14}{97}\langle 4, -9 \rangle=\langle -\frac{56}{97}, \frac{126}{97} \rangle=-\frac{56}{97}i+\frac{126}{97}j$
(c) Clearly $\vec u_1=proj_v\vec u=-\frac{56}{97}i+\frac{126}{97}j$, and $\vec u_2=\vec u-\vec u_1=\vec u-proj_v\vec u=(1+\frac{56}{97})i+(2-\frac{126}{97})j=\frac{153}{97}i+\frac{68}{97}j$
