1. Let A= (-6,-4), B=(3,2) and c=(6,4)

(a) These points lie on a line through the origin. Find its slope.

2/3

(b) Letube the vector whose components are the x-coordinates of A,B and C, and letvbe the vector whose components are the y- coordinates of A,B,and C. Show thatuandvpoint in the same direction.

u = (-6,3,6) v= (-4,2,4)

3u = 2v

(c) Explain why the ratio ofl v ltol ul is the slope you found in part (a).

36/81 = 2/3

I can see that the ratio of the magnitudes of the vectors are the same as the slope in (a) but I am not sure what the problem wants me to be seeing.....

(d) What would the vectorsuandvhave looked like if A,B,and C had not been collinear with the origin? Hmmm... They would be skew vectors?

2. Let A= (-3,-2) B= (-1,-1) C= (4,3), u=[-3,-1,4] and v=[-2,-1,3]

(a) Show thatuandvpoint in different directions. Let w be the vector that results when v is projected onto u. Show that w is approximately [-2.19,-0.73,2.92]

cosx= u.v/uu therefore 19/26 = 0.73

0.73[-3, -1,4] = [ -2.19, -0.73, 2.92]

(b)Make a scatter plot. Verify that A,B, and C are not collinear. Notice that the x-coordinates of these points are the components ofuand the y coordinates are the components ofv, suggesting thatuandvare like lists in your calculator.

I made a scatter plot and I see that the x,y coordinates are the components of each vector but what does like lists in your calculator mean?

(c) Verify that the points A' = (-3,-2.19) B' = (-1,-0.73) and C' =(4,2.92) lie on a line that goes through the origin. Notice that the y-coordinates of A',B' and C' appear as the components ofw, and that they are proportional to the components ofu.

I understand this part.

(d) You calculated w by first finding that it is m times as long asu, where m isuv/uu.Notice that m is also the slope of the the line through A',B' and C'. Now use your calculator to find an equation for the so called regression line(or LinReg)for the data points A,B,and C. The slope should look familiar.

This problem threw me off track because now I have to study

(Regression and Correlation relying mostly on google).

I am slowly starting to see the relation between the regression line and vectors but this will take some time. Can I get some insights, if possible?

(b) Verify that the vectorr=v-wis perpendicular tou, then explain why this should have been expected. Is is customary to callra residual vector, because it is really just a list of residuals. Review the meaning of this data analysis term if you need to.

r is perpendicular to u because the vector v-w is a projection of v onto u.

But why is it called a residual vector?

(c) The regression line is sometimes called theleast squares line of best fit, becausewwas chosen to makeras short as possible. Explain this terminology. You will need to refer to the Pythagorean formula for calculating the length of a vector.

I think this is related to the error sum of squares and I think I understand SSE but i am not sure how to relate this to vectors.

I find this problem absolutely fascinating. Only if I could understand more....

I will be studying this problem for the next few weeks so any help will be really appreciated.

Thanks.