Scalars and Vectors Study Guide
We start this lesson by briefly considering a reference frame and the coordinates of a point-like object relative to it. Next, we make a distinction between scalar and vector quantities, and present examples of both. At the end of the chapter, we take a quick look at the properties of vectors.
In order to be able to locate a point-like object on a line, on a surface, or in space, we need to choose a reference position called an origin and then give directions regarding how to get to the object starting from the origin.
A reference frame consists of an origin through which one or more axes specifying some given directions are passing. Coordinates are sets of numbers that uniquely specify the position of a point-like object with respect to this certain reference frame. These coordinates measure certain properties of the path from the origin to the object when we follow these axes according to some given instructions
The simplest case is the one-dimensional motion, along a straight line. In this case, the line is also the only reference axis we need. In order to completely specify the position of an object, we have to choose a fixed point as our origin, and then find the distance from the object to the origin using some convenient units. Because our object may be either to the left or to the right of the origin, we consider by convention that the distance is positive if the object lies to the right of the origin and negative if it lies to the left. This distance and the corresponding sign is what we call the coordinate of the object with respect to our frame of reference. The distance between two points in this case is just the difference between their coordinates. In Figure 2.1, the distance from P to Q is PQ; then PQ = [4.5 – (–2.5)] = 7 units.
In the two-dimensional case, a frame of reference is specified by an origin and a set of two axes passing through that origin. To get to the position of a given object at point P, we may say: Start from the origin and follow x-axis for a length of three units; then follow the direction of y-axis (that is, follow a parallel to the y-axis) for another four units. That is, our object has an x-coordinate of 3 and a y-coordinate of 4 (see Figure 2.2).
If the two axes are perpendicular to each other, we can find the distance from the origin to point P by applying Pythagoras' theorem:
d2 = x2 = y2
d2 = 32 = 42
d2 = 25
d = 5
The most widely known two-dimensional (2-D) system of coordinates or reference frames is the Cartesian one, when the two axes are perpendicular to each other. Given any point P, we draw parallel lines to the axes. The distance from where these lines intercept the axes to the origin are the two coordinates (x,y) of point P. This way of labeling points is known as the Cartesian system, in honor of the French mathematician and philosopher Renee Descartes (1596-1650), who was the first to think of this system.
A quick reference to Pythagoras's theorem gives the distance between two points with coordinates (x1, y1) and (x2, y2) as:
d2 = (x1 – x2)2 + (y1 – y2)2
Another widely used way of specifying a point's position on a plane surface is by using its so-called polar coordinates (r, θ) with r > 0.
One coordinate is the length r of the segment OP from the origin of the reference frame to the point, and the other is the angle θ that this segment OP makes with a reference axis Ox. This angle θ gives the direction to the point P, and r gives the distance to the origin. See Figure 2.3.
If the origin and the x-axis of the two representations are the same, then we can relate these sets of coordinates by the following relations:
x = r · cos θ
y = r · sin θ
r2 = x2 + y2
Depending on the symmetry of the problem we try to solve, it may be easier to use one or the other of these two representations.
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