A straight line is infinitely long.
(what is shown here
is actually a RAY- it may extend infinitely on one side, but has at
least one end)It extends infinitely onto both sides. What then could be the parameters to define a line, and distinguish it from others?
A line may be defined in many systems of calculus, but the one that is most popular is the Cartesian System. This system assumes an axial type of arrangement - with X, Y, Z as the Axes in case of a 3D arrangement:
The coordinate system assumes perpendicular axes all meeting at the point called ORIGIN, and mentioned O. In the 3 Dimension, this means 3 perpendicular axes - X, Y, Z. In case of more than 3 axes, you may name them anything, but please don't ask me to help you visualize 4 or more axes perpendicular to each other!
Each point in this kind of system of classifying the space around us, is uniquely mentioned through an n-tuple. For 3 D, this converts to a 3 tuple - (x, y, z). Thus, (1, 3, 2) is one point in this 3 D system, and (0, 2, 1.5) is another.
For the 2-Dimension, considering No Loss of Generality(we will discuss this in a separate lesson), it is the X-Y Plane that is generally considered.
Now, the straight line is generally seen in 2 Dimension, which is why we will focus towards only those parameters that are required in 2 Dimension.
A straight line in 2-D could be either of the following type:
- Parallel to the X Axis
- Parallel to the Y Axis
- Any other
Now, how do we distinguish one straight line in 2-D from another? Reiterating, a straight line is infinitely long on both ends. Thus, if we can confirm at least 2 points that it passes through, we are sure that no other line can repeat this achievement of the first line. If another line is to pass through the 2 same points, IT MUST SUPERIMPOSE on the first line, essentially making it the same line - TRY IT FOR YOURSELF USING A PEN & PAPER.
Alternatively, imagine that I knew just 1 point that this straight line is passing through, and instead of knowing another point, I just knew what is the inclination-angle with the X Axis, favoritely called the SLOPE. Do you think another straight line could repeat the same feat? Again, try it for yourself.
Also, if line is of the 3rd type as mentioned above, it is bound to cut both, the X and the Y, axes at some point or the other. These points are called the X and the Y intercepts. In common mathematical usage, it is the Y Intercept that is generally considered.
(the Y Intercept in this case is 2, since the straight line cuts Y Axis at 2 ordinate)
So what comes out is the following. In a 2-D, in order to uniquely define a straight line, we just need 2 parameters.
These are the parameters that could alternatively be used in pairs to define a straight line:
- 1st Point
- 2nd Point
- Slope
- Intercept - particularly the one
on the Y Axis (mostly considered instead of the X Axis)
Caution: Do not consider 1 point (x1, y1) as 2 separate variables - this together is one piece of information. The x1 is TOGETHER with y1, a ONE POINT.
With the knowledge of any of the above 2, we can construct a straight line in 2-D uniquely!
Hope this clears the basics. For numerical questions, and further detail, visit the M 102.
Thanks,
Amit
