A practical on ML basics

to refresh some knowledge.

Geometry

1. Given the two points $(4,1)$ and $(-2,3)$, give an equation for the line through these two points. The equation should have the form:

$$w_0+w_1{x}_1+w_2{x}_2=0$$

There’s the line equation given by two points:

$$\frac{x-x_A}{x_B-x_A}=\frac{y-y_A}{y_B-y_A}$$

so after I plug in the values, I get:

$$\begin{array}{rl}\frac{x-4}{-2-4}& =\frac{y-1}{3-1}\\ 2x-8& =-6y+6\\ x+3y-7& =0\end{array}$$

If we substitute, the solution would be $w=\left(\begin{array}{c}{w}_0\\ w_1\\ w_2\end{array}\right)=\left(\begin{array}{c}-7\\ 1\\ 3\end{array}\right)$

2. Make a plot of the line and the vector $(w_1,w_2)$. The direction of $(w_1,w_2)$ is called the positive side of the line.

3. Consider the point $(x_1,x_2)=(4,3)$. Is this point on the positive side or negative side of the line? Also compute $w_0+w_1{x}_1+w_2{x}_2$ for this point.

I only need to plug the point into the line equation and see if it’s positive or negative.

$-7+4+9=6>0$ ⇒ on the positive side.

4. Show that the distance of the point $(x_1,x_2)=(4,3)$ to the line is equal to:

$$\frac{|w_0+w_{1}x1+w_2{x}_2|}{||(w_1,w_2)||}$$

Distance from a point to a line is effectiv