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To find the magnitude and direction of the acclleration for the uniform circular motion,we consider the below figure enter image description here

Where in particle $p$ moves at constant speed $v$ around a circle of radius $r$.At the instant shown,$p$ has coordinates $x_{p}$ and $y_{p}$. since velocity $\vec{v}$ of a moving particle is always tangent to the particle's path at the particle's position.In the above figure that means $\vec{v}$ is perpendicular to a radius $r$ drawn to the particle's position.

How does the angle $\theta$ that $\vec{v}$ makes with a verticle at $p$ equals the angle $\theta$ that radius $r$ makes with the $x$-axis?

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The linear pair axiom does the job here.

Note that, the line marked $y_p$ is a straight line.

Linear Pair Axioms:

  1. If a ray stands on line, then the sum of two adjacent angles so formed is $180^\circ$.
  2. If the sum of two adjacent angles is $180^\circ$, then the non-common arms of the angles form a line.

Also, note that sum of the interior angles in a triangle is $180 ^\circ$.

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Oh ya..High school geometry! –  alok Jan 31 '12 at 20:04

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