Why are I, J, and K used for unit vectors?

[abrowaqas]

Why Unit vectors are represented by I, J and K ?

Why do unit vectors I, J and K have no units? are the unit vectors in the cylindrical and spherical coordinates system constant vectors?

 

[HallsofIvy]

Why Unit vectors are represented by I, J and K ?

Why not? They are three consecutive letters of the alphabet- and x, y, and z were already taken. (However, I feel I should point out that the unit vectors in the direction of the coordinate axes are NOT represented by "I, J and K", they are represented by i, j, and k. Or, better, by $\vec{i}$, $\vec{j}$, and $\vec{k}$.)

Why do unit vectors I, J and K have no units? are the unit vectors in the cylindrical and spherical coordinates system constant vectors?

If, in a particular application, your x, y, and z variables are in a measured in particular units, then those are the units of the vectors.

I'm not sure what you mean "unit vectors in the cylindrical and spherical coordinates system". No matter what coordinate system you use, you could still use i, j, and k, pointing in the direction of the x, y, and z axes. Those will be constant. Or you could use unit vectors pointing in the direction of r (or $\rho$) and $\theta$ (and $\phi$) at each point which would have constant length (1) but variable direction.

 

[Meir Achuz]

Why Unit vectors are represented by I, J and K ?
Why do unit vectors I, J and K have no units? are the unit vectors in the cylindrical and spherical coordinates system constant vectors?

1. Why not? Whatever they are called, you could ask why?
$${\hat i},{\hat j},{\hat k}$$ are the unit vectors for Cartesian coordinates only. They are constant vectors.
2. Unit vectors have no units because they just signify direction.
4. The unit vectors for cylindrical and spherical coordinates (with other names) are not constant vectors.

 

[fluidistic]

4. The unit vectors for cylindrical and spherical coordinates (with other names) are not constant vectors.

Except the k unit vector of the cylindrical coordinates system which correspond to the k unit vector of Cartesian coordinates system.

 

[homology]

They come from the quaternions 1, i, j, and k. After the complex numbers (with units 1 and i) had some success as candidates for vectors there was a search for a 3D version of the complex numbers. Hamilton worked on this (among others) and was successful in discovering a 4D version (there is no 3D version) and he took the 1 and i from complex numbers and just labeled the two new complex units j and k (in order). j and k are imaginary just as i is.

$$i^2=j^2=k^2=-1$$ and $$ijk=-1$$ were the defining relations of this new structure he'd discovered. A general quaternion is expressed as a+bi+cj+dk and you can use the two properties given above to figure out what you get if you multiply them.

If you start with $$a_0+a_1i+a_2j+a_3k$$ and $$b_0+b_1i+b_2j+b_3k$$ and take their product (a-stuff times b-stuff) you'll get:

$$a_0b_0-\vec{a}\cdot \vec{b}+a_0\vec{b}+b_0\vec{a}+\vec{a}\times\vec{b}$$

where the vector part of a quaternion $$a+bi+cj+dk$$ is $$bi+cj+dk$$.

Hamilton, so the story goes, was a horrible writer and after he wrote up his treatise on the quaternions no one could read it or understand it. A few brave souls tried and in the end (I've forgotten some of the middle story) Gibbs and Heaviside salvaged something useful out of it by separating quaternions into scalar and vector parts. This analysis that they derived from quaternions is the basis for the vector analysis we use today, thus the i, j, and k which have their original roots in complex numbers.

 

[abrowaqas]

fantastic reason given by homology..

but you didn;t answer the second part of my question.

meir please give reasons.thanks hallsofivy you give good explanation..

 

[homology]

The unit vectors have constant magnitude however in some coordinate systems their directions change.

Consider the following:
$$\hat{r}=\cos\theta\hat{i}+\sin\theta\hat{j}$$
$$<br /> \hat{\theta}=-\sin\theta\hat{i}+\cos\theta\hat{j}$$

If you want to describe a particle moving in the plane and you have its position expressed using $$\hat{r},\hat{\theta}$$ and you differentiate with respect to time you can see (and calculate) that $$\dot{\hat{r}}\neq 0$$ and $$\dot{\hat{\theta}}\neq 0$$. Similiarly for double dots.

You can use these relations to do things like write Newton's second law in polar coordinates.

 

[abrowaqas]

thanks homology...

i got it.

you give nice explanation..