# Math Help - Vector Addition Problem

1. ## Vector Addition Problem

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:

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

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

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

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

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

Then, express your answer as the sum of the vectors.

Post what you get!

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

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

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

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

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