Working with vectors I'm still learning how to work with vectors and ran to this question...
Given vector: $V = 3i + 4j$ and for vector $F = 9i + 12j$
a)  Find the component of F parallel to V
b)  the component of F perpendicular to V
c) The work, W, done by force F through displacement
For question a)
i know that to vectors a parallel when $\lambda(V) = F$ but i'm not so sure how use it with equations i'm just used to working with the $(x,y,z)$ vectors
For question b)
I think to be perpendicular the projection of v - projection f must be 0. 
For question c)
i got nothing
 A: The component of f parallel to $v$ is the projection of $4$ onto $v$, $\text{proj}_v(f)$.  The component of $f$ perpendicular to $v$ is $f-\text{proj}_v(f)$.
Try drawing vectors in $\mathbb{R}^2$ to see this.  Draw two vectors $f$ and $v$.  Then draw the projection of $f$ onto $v$.
A: For a)
v = 3i + 4j = <3,4>
f = 9i + 12j = <9,12>
f = v 
when f = 3v
Hence since  vectors are separated by a factor of 3  from each other then they are parallel
For b)
By definition two vectors are parallel when the dot product is zero $a . b = x1x2 + y2y3 = 0 $  
$$a.b = 0$$ 
but the dot product is ... $f.v = 3*9 + 4+12 = 27+48 = 75$ which is not zero hence they are no perpendicular
For c) 
the work is the dot product $w = f.v = |f|*|v|*cos(\theta)$
A: Note that $\textbf{F} = 3\textbf{v}$ so it's clear that $\textbf{F}$ is parallel to $\textbf{v}$ or $\textrm{proj}_\textbf{v} \textbf{F} = \textbf{F}$. It follows that the perpendicular component of $\textbf{F}$ is $\textbf{0}$. 
For the last question, work is $W = \int \textbf{F}\cdot d\textbf{r}$ where $\textbf{r}$ is displacement. You didn't specify what the displacement is. $\textbf{v}$ is usually a notation for velocity. If $\textbf{v}$ is displacement then
$$W = \textbf{F} \cdot \textbf{v} = (3\textbf{i}+4\textbf{j})\cdot(9\textbf{i}+12\textbf{j}) = 3\cdot 9 + 4 \cdot 12 = 75 $$
