To help visualize this, consider the set of all scalar multiples m. They form a line...

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To help visualize this, consider the set of all scalar multiples m. They form a line through the origin. Here's a picture*: y For any one of these vectors m along the line, the vector - m can be visualized as going from the tip of m to the tip of y. Here's a new picture showing that: y mx - m i-mi mx mx So finding the m for which - m is as short as possible is the same as finding the m for which the tip of m is as close as possible to the tip of y. Finally, we get to a question: 8. Using the picture to help you, what point on the line is closest to the tip of y? (Out of all points on the line, not just the few shown in the picture.) Write down a simple formula that you could use to find m. Or in other words: the point you just described is what scalar multiple of x? 9. Let's actually use this method on some data. Consider the following data points: x Y 6 5 1 1 1 2 4 2 3 3 25 3 5 2 44 46 1 1 2 3 4 (a) Write down the vectors x and y. (Recall that I just consists of all the x, values, and consists of all the y values.) (b) Find the best (least squares) fit of the form y = mx for these data points. That is, find the value of m using the method you figured out in the previous problem. You might want to check that your answer is in the right ballpark by drawing the resulting line on the graph above, and seeing if it looks like it fits the data well. To help visualize this, consider the set of all scalar multiples m. They form a line through the origin. Here's a picture*: y For any one of these vectors m along the line, the vector - m can be visualized as going from the tip of m to the tip of y. Here's a new picture showing that: y mx - m i-mi mx mx So finding the m for which - m is as short as possible is the same as finding the m for which the tip of m is as close as possible to the tip of y. Finally, we get to a question: 8. Using the picture to help you, what point on the line is closest to the tip of y? (Out of all points on the line, not just the few shown in the picture.) Write down a simple formula that you could use to find m. Or in other words: the point you just described is what scalar multiple of x? 9. Let's actually use this method on some data. Consider the following data points: x Y 6 5 1 1 1 2 4 2 3 3 25 3 5 2 44 46 1 1 2 3 4 (a) Write down the vectors x and y. (Recall that I just consists of all the x, values, and consists of all the y values.) (b) Find the best (least squares) fit of the form y = mx for these data points. That is, find the value of m using the method you figured out in the previous problem. You might want to check that your answer is in the right ballpark by drawing the resulting line on the graph above, and seeing if it looks like it fits the data well.