What comes after (i hat j hat k hat)

[billturner90]

hello persons of the forum i am new here. I am in my second year of college and i recently became a bit more interested in math after finishing my calculus series. The 3 unit vectors used (i,j,k) are all orthogonal if i am correct. I was wondering if once we look into the 4-D space would the next unit vector also be orthogonal to the 3 previous unit vectors? is this where non euclidian geometry comes into play?

 

[Char. Limit]

It's not quite "non-Euclidean", but I think the next unit vector would be $$\hat{l}$$, for the w-axis.

 

[billturner90]

thanks for the response but maybe you misunderstood my question. I was wondering if, as we move up in the dimensions, each new unit vector would be orthogonal to all the others. I guess i feel curious about this because it is not possible to visualize 4-D space, and i don't understand how this new unit vector would look.

 

[Mark44]

To the best of my knowledge, there is no name for that unit vector. Three-dimensional space is very commonly used in physics problems, so there are special names for the unit vectors in that space.

 

[Fredrik]

The 3 unit vectors used (i,j,k) are all orthogonal if i am correct. I was wondering if once we look into the 4-D space would the next unit vector also be orthogonal to the 3 previous unit vectors?

It is if we want it to be. The standard basis for the vector space $\mathbb R^4$ is

$$\vec e_1=(1,0,0,0)$$
$$\vec e_2=(0,1,0,0)$$
$$\vec e_3=(0,0,1,0)$$
$$\vec e_4=(0,0,0,1)$$

These vectors are orthogonal with respect to the standard inner product, defined by

$$\vec x\cdot\vec y=x_1y_1+x_2y_2+x_3y_3+x_4y_4$$

 

[HallsofIvy]

thanks for the response but maybe you misunderstood my question. I was wondering if, as we move up in the dimensions, each new unit vector would be orthogonal to all the others. I guess i feel curious about this because it is not possible to visualize 4-D space, and i don't understand how this new unit vector would look.

It is not necessary that basis vectors be orthogonal to one another but calculations are much simpler if they are, so, yes, the "standard" basis for $R^n$ has all basis vectors orthogonal to one another (and of unit length).

 

[D H]

thanks for the response but maybe you misunderstood my question. I was wondering if, as we move up in the dimensions, each new unit vector would be orthogonal to all the others. I guess i feel curious about this because it is not possible to visualize 4-D space, and i don't understand how this new unit vector would look.

It certainly is possible to visualize 4D space. The same techniques used to represent a 3D object on a 2D piece of paper can be applied to higher dimensions. Use a search engine to search for images of a tesseract and you will find lots of such visualizations. It is a bit difficult to wrap your mind, but it is not impossible.Regarding the names i hat, j hat, k hat: Those are far from the only names used to describe the canonical R³ unit vectors (1,0,0), (0,1,0), and (0,0,1). You will also see these vectors identified as x hat, y hat, z hat, as or e₁, e₂, e₃, and so on.

The names $$\hat{\imath}$$, $$\hat{\jmath}$$, and $$\hat k$$ are pretty much specialized to 3D space. Those names stems from the quaternionic origin of modern vector analysis. Hamilton envisioned an extension to the complex numbers in which i²=j²=k²=ijk=-1. Some physicists very much liked Hamilton's quaternions, others very much didn't like them. Those who didn't like the quaternions did see their utility in describing our physical world. The vectorialists developed our modern 3D vector analysis by combining the "useful content" of Hamilton's quaternions with the (erroneously) discarded works of Hermann Grassmann. An intellectual war between the quaternionists and vectorialists ensued during the latter part of the 19th century into the early 20th century. The vectorialists won that battle in the sense that we now largely use vectors rather than quaternions to describe the 3D world.

The names $$\hat e_1$$, $$\hat e_2$$, and $$\hat e_3$$ comes from German mathematicians. Those German mathematicians also saw the utility of Hamilton's quaternions and of the (largely) English-speaking vectorialists. They also saw that Grassmann's work was of even greater value than that used by the vectorialists. Vector analysis is useful far beyond the apparently 3D world in which we appear to exist. One obvious way to generalize the concept of the 3D unit vectors to some N-dimensional space is to use one symbol to designate a unit vector, with indices to indicate different unit vectors. Since this work was done largely by German mathematicians, the symbol e used to designate a generic unit vector stems from the German word einheitsvektor ("unit vector").

 

[discrete*]

Just as a follow up to D H's excellent and intriguing post, check out the wiki entry for tesseract -- it's really interesting and may clear up some of the OPs original concerns.