# properties of the zero vector

• Feb 21st 2008, 05:14 AM
billym
properties of the zero vector
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> ?
• Feb 21st 2008, 08:15 AM
Soroban
Hello, billym!

Interesting . . . "Zero" always raises strange questions.

Quote:

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?

There is always a difference between a point and a vector.

The point might be $\displaystyle (2,3)$ and the zero vector is: .$\displaystyle \langle 0,0\rangle$

Quote:

Can there be a polar representation of a single point?
If so, would it be $\displaystyle \langle 0\rangle$ ?

If you referring to the pole (origin), the polar coordinates are: .$\displaystyle (0,\theta)$
. . where the 'magnitude' is 0 $\displaystyle (r=0)$ and $\displaystyle \theta$ can be any angle.

Other points have coordinates such as: .$\displaystyle \left(2,\,\frac{\pi}{6}\right)$

• Feb 21st 2008, 09:06 AM
billym
I'm still confused... what would be the polar representation of of a point with coordinates (-3,4)?
• Feb 21st 2008, 09:12 AM
bobak
Quote:

Originally Posted by billym
I'm still confused... what would be the polar representation of of a point with coordinates (-3,4)?

$\displaystyle x = r \cos \theta$
$\displaystyle y = r \sin \theta$

If you want to switch form Cartesian to Polar

note that $\displaystyle x^2 + y^2 = r^2$
• Feb 21st 2008, 09:23 AM
billym
In terms of direction, do you treat the point as if it were a position vector, and then just state the magnitude as zero?
• Feb 21st 2008, 09:35 AM
Plato
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: $\displaystyle \left( { - 3,4} \right) \equiv \left( {5,\pi + \arctan \left( {\frac{{ - 4}}{3}} \right)} \right)$.
• Feb 21st 2008, 09:55 AM
billym
but I'm trying to find the polar representation of just a point (-3,4), how could a point have magnitude of 5?
• Feb 21st 2008, 10:15 AM
bobak
Quote:

Originally Posted by billym
but I'm trying to find the polar representation of just a point (-3,4), how could a point have magnitude of 5?

Spend some time reading this, Polar coordinate system - Wikipedia, the free encyclopedia