Is a point "P" on a plane considered a zero vector, PP, or is there a difference between a single point and a zero vector?
Can there be a polar representation of a single point? If so, would it be <0> ?
Hello, billym!
Interesting . . . "Zero" always raises strange questions.
There is always a difference between a point and a vector.Is a point $\displaystyle P$ on a plane considered a zero vector, $\displaystyle \overrightarrow{PP}$
or is there a difference between a single point and a zero vector?
The point might be $\displaystyle (2,3)$ and the zero vector is: .$\displaystyle \langle 0,0\rangle$
If you referring to the pole (origin), the polar coordinates are: .$\displaystyle (0,\theta)$Can there be a polar representation of a single point?
If so, would it be $\displaystyle \langle 0\rangle$ ?
. . where the 'magnitude' is 0 $\displaystyle (r=0)$ and $\displaystyle \theta$ can be any angle.
Other points have coordinates such as: .$\displaystyle \left(2,\,\frac{\pi}{6}\right)$
It would be quite fair if someone argued that yours is a philosophical rather than a mathematical question.
Vectors are hybrid objects in that a vector has both length and direction.
Here is a quotation from Melvin Hausner, a geometer at NYU, “The vector $\displaystyle \overrightarrow {PP} $ will be designated by 0 (read: zero, or the zero vector, depending on how fussy you are.) Thus $\displaystyle \overrightarrow {PA} = 0$ is a way of writing $\displaystyle P = A$.”
To answer your other question: $\displaystyle \left( { - 3,4} \right) \equiv \left( {5,\pi + \arctan \left( {\frac{{ - 4}}{3}} \right)} \right)$.
Spend some time reading this, Polar coordinate system - Wikipedia, the free encyclopedia