Answer
$\lt \frac{20}{17},\frac{-5}{17} \gt$
Work Step by Step
By definition: $proj_ab=\frac{ab}{|a|^2}a$
$a = \lt 1,4 \gt, b= \lt 2,3 \gt$
$a \cdot b= 1 (2)+4 (3)=14$
$|a|= \sqrt {17}$
and $|a|^2=17$
$proj_ab=\frac{ab}{|a|^2}a$
$=\frac{14}{17}\lt 1,4 \gt$
$=\lt \frac{14}{17},\frac{56}{17} \gt$
$orth_a{b}=b-proj_ab$
$=\lt 2,3 \gt-\lt \frac{14}{17},\frac{56}{17} \gt$
$=\lt 2-\frac{14}{17},3-\frac{56}{17} \gt$
$=\lt \frac{20}{17},\frac{-5}{17} \gt$
