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For some constant vector $\vec{A}$, and for $\vec{r}$ the position vector with respect to the origin, describe the loci of points defined by the following equations:

a) $(\vec{r} - \vec{A}) \cdot \vec{r} = 0$

b) $(\vec{r} - \vec{A}) \cdot \vec{A} = 0$

I'm not sure how to interpret these. I should be able to look at these and decide that they are circles or whatever have you, but I can't seem to come up with anything that makes sense. I have tried writing out the terms, but nothing enlightening has come to me.

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2 Answers

up vote 2 down vote accepted

If geometrical intuition fails, you can always go with plain old algebra. If $\vec{A}=(a,b)$ and $\vec{r}=(x,y)$, then your two equations write out as:

$$(x-a)x+(y-b)y=0$$

$$(x-a)a+(y-b)b=0$$

A little tampering with those equations should lead you to recognizable forms of the equations of the loci you are after.

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I think the easiest way to imagine this is if you picture $A$ as representing a point. Then $A, r, r-A$ form a triangle, and you should be able to envision the geometries that correspond to these two conditions. Remember that if the dot product of two vectors is zero, then those vectors must be perpendicular.

Edit: you can also use \cdot (for "centered dot") to express dot products.

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