Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 07

MATH152 April 2013

  •  QA 1  •  QA 2  •  QA 3  •  QA 4  •  QA 5  •  QA 6  •  QA 7  •  QA 8  •  QA 9  •  QA 10  •  QA 11  •  QA 12  •  QA 13  •  QA 14  •  QA 15  •  QA 16  •  QA 17  •  QA 18  •  QA 19  •  QA 20  •  QA 21  •  QA 22  •  QA 23  •  QA 24  •  QA 25  •  QA 26  •  QA 27  •  QA 28  •  QA 29  •  QA 30  •  QB 1(a)  •  QB 1(b)  •  QB 1(c)  •  QB 1(d)  •  QB 2(a)  •  QB 2(b)  •  QB 2(c)  •  QB 2(d)  •  QB 3(a)  •  QB 3(b)  •  QB 3(c)  •  QB 4(a)  •  QB 4(b)  •  QB 4(c)  •  QB 5(a)  •  QB 5(b)  •  QB 5(c)  •  QB 6(a)  •  QB 6(b)  •  QB 6(c)  •  

   Other MATH152 Exams

  •  April 2016  •  April 2015  •  April 2014  •  April 2022  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2017  •  

Question A 07
Is it possible for the intersection between two planes in $\mathbb {R} ^{3}$ with non-parallel normal vectors to contain two linearly independent vectors?

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!

Hint
How many possible ways can two planes intersect?

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. 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. No it is not possible for two planes to intersect in two linearly independent lines. Two planes in $\mathbb {R} ^{3}$ intersect in either a line, a plane (which is generated by two linearly independent vectors) or not at all. If they intersect in a plane or not at all, then their normal vectors are parallel. Otherwise they intersect in a line which cannot contain two parallel vectors. For a more technical explanation, consider a point <x,y,z> on both planes. Let the first plane be given by $\displaystyle {}n_{1}x+n_{2}y+n_{3}z=a$ and the second plane by $\displaystyle {}m_{1}x+m_{2}y+m_{3}z=b.$ The normal vectors are not parallel and so $<n_{1},n_{2},n_{3}>$ and $<m_{1},m_{2},m_{3}>$ are linearly independent. In a linear system, we can write this as ${\begin{bmatrix}n_{1}&n_{2}&n_{3}\\m_{1}&m_{2}&m_{3}\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}a\\b\end{bmatrix}}.$ Here we have two equations with three unknowns and this tells us that there is at least one free variable. Since the normal vectors are linearly independent, then there is only one free variable. Geometrically, one free variable corresponds to a line in space and so there is only one line which could contain the points of intersection of the two planes.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Linear independence and bases, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag