Vector orthogonal to U Let y=(3,5) and u=(6,2). 
Write y as the sum of a vector in Span{u} and a vector orthogonal to u.
If someone could do this problem as an example, it would be great.
 A: $y=u_\perp + u_\parallel$
$u_\perp = (y - u_\parallel)$ and 
$u_\parallel = proj_u y = \frac{y\cdot u}{y \cdot y} y$. 
The rest is just plugging into these equations.
A: Since these are vectors in the $xy$-plane, you can also approach it this way.  A vector $\vec{v}$ orthogonal to $\vec{u} = < 6, 2 >$ must produce the dot product $\vec{u} \cdot \vec{v} = 0$.  The easiest way to accomplish this is to choose something like $\vec{v} = < 2, -6 >$ . (Note that this is related to the fact that the product of the slopes of perpendicular lines is $-1$ .)
We want $\vec{y} = < 3 , 5 >$ then to be some linear combination of $\vec{u}$ and $\vec{v}$, that is,
$$< 3 , 5 > =  a < 6 , 2 > + b < 2 , -6 > .$$
You would now solve for $a$ and $b$ . However, an infinite number of choices are permissible, since we could have multiplied $\vec{v}$ by any non-zero number to make it any length we wished (or even reversed its direction).  So you could pick either $a$ or $b$ to be 1 (or really any non-zero value you like), and solve for the remaining coefficient. The vector in Span(u) is $a\vec{u}$ and your orthogonal vector will be $\vec{w} = b \vec{v}$.
