# Thread: Vector projection

1. ## Vector projection

u= (((3i + j) dotted with (i + j)) * (i +j)) / (|i = j|^2). I tried to simplify to
u = (3i + j)*(i + j) / (i^2 +j^2) and from here i don't know how to simplify further. Apparently it is simplified to 4/2 (i + j) = 2i + 2j.
The initial question states, express the vector 3i + j as a su of vectors u + v, where u is parallel to vector i + j and v is perpendicular to u. I am interested in the vector projection solution.

Help would be nice!

2. Originally Posted by Zamzen
express the vector 3i + j as a sum of vectors u + v, where u is parallel to vector i + j and v is perpendicular to u. I am interested in the vector projection solution.
Your post is very difficult to read.
So I will answer the question quoted above.

Suppose that $\displaystyle \vec{a}~\&~\vec{b}$ are vectors.
We can write $\displaystyle \vec{a}$ as the sum $\displaystyle \vec{a} =\vec{b}_{||} +\vec{b}_{\|}$ where $\displaystyle \vec{b}_{\|}$ is parallel to $\displaystyle \vec{b}$ and $\displaystyle \vec{b}_{\bot}$ is perpendicular to $\displaystyle \vec{b}$.

$\displaystyle \vec{b}_{\|}=\dfrac{\vec{a}\cdot\vec{b}}{\vec{b}\c dot\vec{b}}~\vec{b}$

$\displaystyle \vec{b}_{\bot}= \vec{a} -\vec{b}_{\|}$