Coordinate Geometry 05: Geometry of a line

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Let us move our study towards a line.

A line is a family of points which satisfy some single equation or relation of every axes

For example, a line AB with relative equation y=x would result in a line that goes 45° from x axis on x-y plane (lines travel in a single plane : either of x-y /y-z / x-z planes)

Let's define the variables which come handy to find complicated cases of equations of lines.

Slope of a line

Slope of a line refers to the inclination of y-component or z-component of a line on its x-component

It gives us the inclined-ness of a line with respect to x axis, however, we measure inclinedness using angle measured by anticlockwise rotation about the intersection point O : in textbook language we call that to be measuring angle from positive x-axis

Equation of line here is y=x as on one unit moves on x-axis, 1 unit is moved on y - axis

Equation of a line is a beautiful way to interconnect the dependency of movement in y and z on the per unit movement  in x-axis.

But not only this... There can be infinite lines in the above example that have same +1unit on movement in y on +1 unit movement in x dependency.

Not only y = x, there is another line y= x + 1 . Now let carefully observe that line y = (x) +1 as follow :

If we carefully look to observe, we find that this line also travels same distance which y=x line made previously.

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With this we conclude that the extra term +1 in y=x+1 is just a location marker. 

Now let us look to a line y=2x

    This new line travels + 2 units on Y axis on left to right movement of +1 unit on X... y=2x + 0 the plus 0 states that it's location exactly is that y= (2x) which passes by cutting y axis on 0 on Y.

Similarly a line y =3x would therefore means y = 3x + 0  for exactly locating that line.

Thus we could establish that equation of a line is formed of two components : slope and locating y-intercept or z-intercept

Y=(Slope)X + y-intercept

Z=(Slope)X +z-intercept

Slope = $\frac{y_2-y_1}{x_2-x_1}$

Complicated Lines

We told you briefly in between that lines move in a single plane, either xy / yz /zx.

Infact, dear friends, we could adjust our plane in a manner that our line falls on that single plane.

But in reality we never invest our time into redefining x-, y- and z- just fore the sake of getting that line in a plane surfaces {2D}

Due to this  that line starts travelling in 3D, 2 planes.

For such lines we define equation in such a general format :

$\mathrm{\frac{{x - x_0}}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}}$