could anyone explain the physical significance of the properties of the VECTOR SPACE.?

Question

we r made to memorize the properties of the VECTOR SPACE . But i want to understand its importance in physical sence. thanks

Answer 1

Its importance is that it gives us a way to represent magnitudes and directions with sets of numbers, and employ mathematical techniques on them, making the problem simpler, rather than abstract like it is to start with.

Your question is very general. There is no one vector space.
The general definition for a vector space is as follows.

In mathematics, a vector space is a collection of objects (called vectors) that may be scaled and added. More formally, a vector space is a set on which two operations, vector addition and scalar multiplication, are defined and satisfy certain natural axioms. Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, science, and engineering.

The most familiar vector spaces are two and three-dimensional Euclidean spaces. Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors which are quantities with a magnitude and a direction, usually depicted as arrows. These vectors may be added together using the parallelogram rule (vector addition) or multiplied by real numbers (scalar multiplication). The behavior of geometric vectors under these operations provides a good intuitive model for the behavior of vectors in more abstract vector spaces, which need not have a geometric interpretation. For example, the set of (real) polynomials forms a vector space.

There are many applications to many branches of the physical sciences, too many to name. It is more important to understand the properties, and to learn vector calculus.

Answer 2

The physical significance is that the spatial dimensions have both magnitude and direction. This is why we say something like "go 2 blocks west and then turn north and go 5 blocks" when giving directions. The 2 and 5 blocks are the magnitudes and the west and north are the directions. It's that simple, nothing more...just magnitude and direction, which make the spatial dimensions vectors.

There are physical ramifications of vector space, however. The easiest one to see is the difference between speed, which is a scalar having only magnitude, and velocity, which is a vector having both magnitude and direction. Here's an example of that difference.

Suppose you look at you odometer on your car just before leaving the house and going on some errands in town. It reads 000000, a brand new car.

When you finish the errands back at home, the odo reads 000060...you drove sixty miles. And, noting your wristwatch as you climb out of the car, it took 3 hours. What was your average speed? That would be D/T = s; where D = 60 mi, T = 3 hours, so your average speed is D/T = 60/3 = 20 mph. Easy.

But here's an even easier question to answer. What was your average velocity after your three-hour shopping spree?

If you answered nada, zilch, niente v = 0 mph, then you can go to the head of the class. And that's the difference. Since speed is a scalar, we don't take displacement (direction) into consideration when averaging speed. But we do when averaging vectors, like velocity, which has both direction and magnitude.

v = 0 because displacement = S = 0. How do we know that? It's because you started from home and ended at home; you ended up where you started from. There was no displacement after driving around for three hours and v = S/T = 0/T = 0.

And that is the physical significance of vector space. There are a lot of math differences as well. In fact, a branch of math called vectors analysis comes from the fact that our spatial dimensions are vectors.