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western50 · Jun 20, 2011, 08:16 AM

So I have these two vectors
X=3 {Cos[\[Pi]/3],Sin[\[Pi]/3]};
Y={Cos[\[Pi]/6],Sin[\[Pi]/6]};
and I want to obtain two other vectors U and V such that obey the following conditions:

U parallel to Y
V perpendicular to Y
X = U+V

I have tried many ways but still can't get the answer for U and V, please help me on this!

ebaines · Jun 20, 2011, 09:49 AM

Let A equal the magnitude of $\vec U$ . Since you know $\vec U$ is parallel to $\vec Y$ , it's operating in the direction $\pi/6$ . Break $\vec U$ it into its x and y components:

$<br /> \vec U= A \cos ( \frac {\pi} 6) \hat x + A \sin (\frac {\pi} 6) \hat y<br />$

The vector that is perpendicular to Y has magnitiude B, and is in the direction of $\vec Y$ plus $\pi/2$ :

$<br /> \vec V= B \cos ( \frac {2 \pi} 3) \hat x + B \sin (\frac {2 \pi} 3) \hat y<br />$

Now add the x components of the $\vec U$ and $\vec V$ vectors together, and set this equal to the x component of your $\vec X$ vector, and do the same for the y components. This will give you two equations in 2 unknowns; solve for A and B.