Source: WikipediaSuch a map (field) is a SCALAR field. Each point on the map has a single denominate number associated with it (a denominate number is simply a number that has units - here, the height could be, say, 2,450 feet). Other examples of scalar fields would be temperature maps, showing the temperature at each location, or air pressure maps.
A VECTOR field, by contrast, is defined by a vector at each position. If you look at a wind speed map, for example, you would have to describe not only the strength of the wind at each location, but also the direction of the wind.
Vectors, as in the image above, are generally represented by arrows; the length of the arrow represents the magnitude of the vector (wind speed) whereas the direction of the arrow represents the direction of the vector.
As you can see, scalars and vectors are very useful in describing things. Scalars are, however, in some ways, merely special cases of vectors - that is, they are vector fields where we're not really interested in direction. For example, you could argue that the topographic (contour) map should have little arrows pointing up, because we are representing magnitude in the direction "up". But because we take that as a given, we can represent altitudes simply as magnitudes.
Lets take these concepts again from a different angle. Suppose I ask you how many houses there are on a street - You might answer "ten". This would be an appropriate answer because it provides a full answer. However, if I ask you how far is it from the first house to the last house, giving me the answer "50" would not be useful, because you might mean "50 yards", "50 feet" etc. Once you have given me this denominate number, you have presented me with a scalar.
But if I then ask you to tell me how to get from the first house to the last house, telling me to walk "50 yards" would not be useful, until you have given me a direction in which to walk. This would then be a vector -"50 yards due south". This is a pretty simple vector. A more complex vector (but still a vector) would be to say "50 yards south, 3 yards west, and 4 yards up". Although we are now dealing with three coordinate axes, we still have one arrow - one direction, with one magnitude.
Scalar and Vector Maths:
Scalars can be treated according to normal maths, for example, 1 mile + 2 mile = 3 miles.
However, Vectors are a little more complicated. Adding vectors, for example, is done by placing the arrows head to tail and then drawing a new vector joining the free ends together. So If I tell you to walk 3 miles south and then 4 miles east, you're going to end up 5 miles in a more or less southeast direction. You have added 3S + 4E = 5 SE.
Vectors can also be (1) "multiplied" in a "dot product", "inner product" or "scalar product" to give a scalar
(n = a unit vector perpendicular to a and b).
Scalars can also be multiplied by vectors to give new vectors (with the same direction). E.g., 5 miles north x 2 = 10 miles north.
In physics, many quantities are either scalars or vectors, and that affects how they can be treated.For example, Mass is a scalar, but acceleration is a vector. The product of the two is a vector; F = M A
(force = mass x acceleration).
Force (vector) [dot product] velocity (vector) = power (scalar)
Force (vector) [dot product] displacement (vector) = power (scalar)
Linear momentum (vector) [cross product] distance vector = angular momentum (vector)
Force (vector) [cross product] distance vector = Torque (vector).
