Vector Addition Problem

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• Nov 12th 2010, 07:23 PM
r2d2
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
• Nov 12th 2010, 09:19 PM
Unknown008
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! (Smile)
• Nov 13th 2010, 04:57 AM
HallsofIvy
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 $\vec{u}$.
• Nov 13th 2010, 06:27 AM
Plato
Here is some different notations.
Suppose that $U~\&~V$ are two non-parallel vectors.
Then $U = V_{||} + V_ \bot$ where $V_ \bot = \frac{{U \cdot V}}{{V \cdot V}}V~\&~U_{||} = U - V_ \bot$.