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

MATH152 April 2013

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[hide] 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!

[show] Hint
How many possible ways can two planes intersect?

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[show] Solution

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.

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