1. The problem statement, all variables and given/known data

Write the vector <1,7> as a sum of two vectors, one parallel to <2,-1> and one perpendicular to <2,-1>

2. Relevant equations
DOt Product

3. The attempt at a solution

I'm confused on where to begin this problem. Should I be using the dot product?

Thanks

2. You can do it like this:

$\displaystyle \displaystyle a \binom{2}{-1} + b \vec{n} = \binom{1}{7}$

Where $\displaystyle \displaystyle \vec{n} = \binom{x}{y}$

To find the perpendicular vector n, use the dot product:

$\displaystyle \displaystyle \binom{2}{-1}\cdot \binom{x}{y} = 0$

Find a value of x and y that fit in this dot product other than zero. Making a quick sketch will confirm your answer.

Post what you get!

3. But you can do it using the dot product.
The projection of vector $\displaystyle \vec{u}$ on $\displaystyle \vec{v}$ has length $\displaystyle |\vec{u}|cos(\theta)$ where $\displaystyle \theta$ is the angle between the two vectors. It is also true that $\displaystyle \vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta)$ so that the projection of $\displaystyle \vec{u}$ on $\displaystyle \vec{v}$ has length $\displaystyle |\vec{u}|cos(\theta)= \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}$.

In order to make that into a vector in the direction of $\displaystyle \vec{v}$ multiply it by the unit vector in the direction of $\displaystyle \vec{v}$:
$\displaystyle \frac{\vec{v}}{|\vec{v}|}$ to get

$\displaystyle \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|^2}\vec{v}$

That is the component of $\displaystyle \vec{u}$ in the direction of $\displaystyle \vec{v}$. To get the conponent perpendicular to that direction, subtract that from $\displaystyle \vec{u}$.

4. Here is some different notations.
Suppose that $\displaystyle U~\&~V$ are two non-parallel vectors.
Then $\displaystyle U = V_{||} + V_ \bot$ where $\displaystyle V_ \bot = \frac{{U \cdot V}}{{V \cdot V}}V~\&~U_{||} = U - V_ \bot$.