MATH152 April 2011
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Question A 03
Given the vectors
Compute the projection of x onto the direction of y.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Consider drawing a line segment to form a right triangle with the two vectors. If you knew the angle between the vectors, how would you get the lengths of the perpendicular sides? How could you determine the angle between the vectors?
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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Consider connecting the two vectors into a right triangle similar the picture to the right. To get the component of onto , we follow like in the picture and take,
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where is the angle between the vectors and and is a unit vector in the direction of . We can rewrite in terms of the dot (scalar) product as,
Therefore, the projection of onto is
where we have recognized that
Computing the dot product we get,
and for the magnitude of ,
Therefore we get that the projection of onto is,