Thanks for your continuing great questions . . . and, unfortunately due to our quite limited resources here, we’re going to have to restrict ourselves to answering questions that are Calculus related. As much as we’d love to help, material involving linear algebra and the “complex world” are beyond our current bandwidth to address.
I will answer your Question #2 here, though:
- In the form y-y1 = m (x-x1) What is the intuition behind getting this form of the line. I mean is there a proof? How do I know that I have to write y-y1 and x-x1 ?
I imagine if I rewrite is as
you’ll recognize the line’s slope. If I gave you two points on a line \left(x_1, y_1\right) and \left(x_2, y_2\right), you’d use that equation to calculate the slope m . . . but if instead I give you the slope m and a point on the line \left(x_1, y_1\right), then you can see how the equation must hold true for any other point \left(x, y\right) on the line. Indeed, the equation you wrote is called the “Point-Slope Form” of a line for just that reason. And you’ll probably end up using this form a line a lot more frequently than the Slope-Intercept form (y = mx + b), including if you have to write the equation of the tangent or normal line to a function’s curve.
I’m sorry we can’t take the time to address your other questions, but trust you can find resources elsewhere – and we remain happy to try to help with any Calculus questions that arise for you!