Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 02

MATH152 April 2013

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Question A 02
Let $a={\begin{bmatrix}-3\\-3\\3\end{bmatrix}}\qquad b={\begin{bmatrix}2\\-3\\-2\end{bmatrix}}\qquad c={\begin{bmatrix}3\\-4\\12\end{bmatrix}}$ What is ${\text{Proj}}_{c}a$?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
The projection ${\text{Proj}}_{c}a$ is the projection of $a$ onto $c$. If you remember the formula for this, you may use it directly. If you don't remember the formula, check the diagram below with $a$, $c$, and ${\text{Proj}}_{c}a$. Use the picture to help you deduce the formula.

Hint 2

Let ${\hat {c}}={\frac {c}{|c|}}$. Using the picture below as a guide for a sketch of what a projection is we can draw the following conclusions. In the image and below we assume that the angle is acute. If the angle is obtuse the final result is the same, and the argument is very similar. $\cos \theta ={\frac {|{\text{Proj}}_{c}a|}{|a|}}$ $\cos \theta ={\frac {a\cdot c}{|a||c|}}$ (by the definition of the dot product) Thus ${\frac {|{\text{Proj}}_{c}a|}{|a|}}={\frac {a\cdot c}{|a||c|}}$ and solving gives $|{\text{Proj}}_{c}a|={\frac {a\cdot c}{|c|}}$ So now we know the length of the desired projection. The direction is c. To make sure that the length of the vector in the final answer is the one calculated above, we choose the unit vector ${\hat {c}}$ instead: ${\text{Proj}}_{c}a=\left({\frac {a\cdot c}{|c|}}\right){\hat {c}}={\frac {a\cdot c}{|c|^{2}}}c$

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We will use the formula, ${\begin{aligned}{\text{Proj}}_{c}a&={\frac {a\cdot c}{|c|^{2}}}c.\end{aligned}}$ Note that $a\cdot c=-9+12+36=39$ and $|c|^{2}=c\cdot c=9+16+144=169.$ Hence, ${\begin{aligned}{\text{Proj}}_{c}a&={\frac {a\cdot c}{|c|^{2}}}c={\frac {39}{169}}c={\frac {3}{13}}{\begin{bmatrix}3\\-4\\12\end{bmatrix}}.\end{aligned}}$

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