I have some confusion over Vectors, Its components and dimensions. Does the number of vector components mean that a vector is in that many dimensions? For e.g. $A$ vector with 4 components has 4 dimensions?
Also, how can a Vector have a fourth dimension? How can we graphically represent vectors with more than 3 dimensions? Its hard for me to visualize such a vector, Can anyone point me to some resource where they explain this graphically and in detail?
In general, a vector in $D$ dimensions will indeed have $D$ components. A vector with $d<D$ components may, however, always be viewed as one in $D$-dimensional space but with $D-d$ of its components equal to zero.
Visualizing higher dimensions is always tough, because it's not something we have any experience with. We only ever see $0$-$3$ dimensional objects. Sometimes I like to just view a higher dimensional vector as an arrow in some abstract space, but really that's not realistic. So most of the time I don't try to visualize the whole vector. I can usually only imagine 3 (or lower) dimensional projections.
The most basic way to imagine 4 dimensions (or more) is as follows:
Picture four straight lines that are all perpendicular to one another. Imagining 3 mutually perpendicular lines is not a challenge, it's the fourth one that's a challenge. The reason for our three dimensional vision is that our retinas are only a 2 dimensional plane, and depth perception adds the 3rd dimension.
One way that I go about imagining a 4 dimensional vector is visualizing a 3 dimensional vector, and attributing some other characteristic as its fourth dimension (usually its color). This also somewhat touches on another part of your question, which is how can a vector have 4 dimensions? A vector doesn't always have to represent a location in space. It can also simply be a set of attributes that form an ordered pair. This is especially common in graphics programming, where color is represented as a vector (each component representing one of the four ARGB channels).
In general relativity (sometimes referred to as Einsteinian coordinates as opposed to Euclidean coordinates), time is a fourth dimension. So the point in time that an object exists and its location in space form a vector in spacetime.
You have to be careful with the notions of components and dimension. Vectors do not have a dimension, although often one speaks of a "n-dimensional vector", which is actually wrong and should be called a vector in a n-dimensional vector space. Vectors are elements of a n-dimensional vector space, and by choosing a basis (consisting of n vectors), one can expand a general vector into a linear combination of the basis vectors, the coefficients being what is called the "components" of the vector. If some of the components are zero, one can embed a vector into a lower dimensional vector space (the hyperplane, in which the vector lies).
Who says that vectors are limited by graphical representations? The math is all the same (except for cross product, which doesn't generalize that well to other dimensions).
In fields like thermodynamics and statistical mechanics one routinely deals with vectors with millions of millions of millions of components or more. We don't bother with a graphical representation, we just know that a vector is simply an ordered tuple of numbers of common dimension (dimension in the sense of units, time/distance etc) and see that there is a surprising amount of use for ordered tuples of numbers millions and millions of elements long... and that these uses have nothing to do with their graphical representation.
protected by Qmechanic♦ Jul 12 '13 at 19:26
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?