Science:Math Exam Resources/Courses/MATH200/April 2012/Question 01 (b)

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MATH200 April 2012

  •  Q1 (a)  •  Q1 (b)  •  Q2 (a)  •  Q2 (b)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q7  •  Q8  •  Q9  •  Q10  •  

   Other MATH200 Exams

  •  April 2012  •  December 2011  •  

Question 01 (b)
Let L be the line which is parallel to the plane $\displaystyle {}2x+y-z=5$ and perpendicular to the line $x=3-t\qquad {}y=1-2t\qquad {}z=3t.$ Find parametric equations for the line L if L passes through a point Q(a,b,c) where a<0, b>0, c>0 and the distances from Q to the xy-plane, xz-plane, and the yz-plane are 2, 3, and 4 respectively.

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
From part (a) we know that the direction vector of the line $\displaystyle L$ is $\displaystyle {\begin{pmatrix}1\\-5\\-3\end{pmatrix}}$ $\displaystyle L=Q+t{\begin{pmatrix}1\\-5\\-3\end{pmatrix}}={\begin{pmatrix}a\\b\\c\end{pmatrix}}+t{\begin{pmatrix}1\\-5\\-3\end{pmatrix}}$

Hint 2

Recall that the distance of a point $\displaystyle P=(p_{1},p_{2},p_{3})$ to the $\displaystyle xy$-plane is the absolute value of the $\displaystyle z$-coordinate $\displaystyle |p_{3}|$, to the $\displaystyle xz$-plane is the absolute value of the $\displaystyle y$-coordinate $\displaystyle |p_{2}|$, to the $\displaystyle yz$-plane is the absolute value of the $\displaystyle x$-coordinate $\displaystyle |p_{1}|$.

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

To find the line to the line $\displaystyle L$, we must find $\displaystyle a,b,c$ such that the point $\displaystyle Q={\begin{pmatrix}a\\b\\c\end{pmatrix}}$ has distance $\displaystyle 2$ to the $\displaystyle xy$-plane. This means $\displaystyle |c|=2$. Since $\displaystyle c>0$, we have $\displaystyle c=2$. The distance from $\displaystyle Q$ to the $\displaystyle xz$-plane is $\displaystyle 3$, hence $\displaystyle |b|=3$. With $\displaystyle b>0$, we have $\displaystyle b=3$. The distance from $\displaystyle Q$ to the $\displaystyle yz$-plane is $\displaystyle 4$, hence $\displaystyle |a|=4$. With $\displaystyle a<0$, we have $\displaystyle a=-4$. Hence $\displaystyle Q={\begin{pmatrix}-4\\3\\2\end{pmatrix}}$ and $\displaystyle L={\begin{pmatrix}-4\\3\\2\end{pmatrix}}+t{\begin{pmatrix}1\\-5\\-3\end{pmatrix}}$.

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