Minimize distance between three points.

[nou-me-na]

Find the value of m such that the sum of the squares of the vertical distances from each of the points (1,1) , (2,2) , and (3,2) to the line y=mx is minimized. Hint: Find the sum as a function of m (no x in the expression) and then minimize it.



Distance equation. d=$\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$



So, our professor made it easier on use and since we have single variable calculus the distance to be found is vertical to the line y=mx. Therefore, the x values will cancel out and we will only be interested in the y values. With this knowledge I set out and tried to set up an equation for the distance formula for each of the values given; but, x values were given in the equation I used for the distance values when I subbed the y value for mx. I'm not sure how to juggle these three equations. How would you proceed with solving this equation. Thank you.

 

[mfb]

Can you show what you did?

I'm not sure how to juggle these three equations.

Add them. The sum of the squared distances should be minimized.

 

[Skins]

How would you express the vertical distance between a given point $$(x_{1},y_{i})$$ and a point on the line y=mx. (hint, what is the vertical distance between the y values ?).