MATH152 April 2012
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) •
[hide]Question 01 (c)
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Let be the plane perpendicular to and through the point P=[1,-1,0]. Let be the plane through the points A=[1,0,0], B=[-1,2,1], C=[0,1,1]. By L, we denote the line of intersection of and .
Determine L in parametric form.
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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?
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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!
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[show]Hint
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We want to determine the line in terms of a single parameter. How does this relate to free variables in a linear system solution?
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[show]Solution 2
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Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
From 1(a) we have that the normal directions to the two planes are
![{\displaystyle {\begin{aligned}\mathbf {n} &=[2,1,-1]\\\mathbf {m} &=[1,1,0]\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/0770368acbde537591afaf93fbfd39f78c56754b)
We know that the line must lie on both planes and therefore the direction of the line must be orthogonal to both normals. Therefore we can find it by taking the cross product,
![{\displaystyle \mathbf {n} \times \mathbf {m} =[2,1,-1]\times [1,1,0]={\textrm {det}}{\begin{pmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\2&1&-1\\1&1&0\end{pmatrix}}=[1,-1,1].}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/3a97e05019320d78b7aef32aa3f2a8e96aa4fe72)
We then must identify a point on the line. To do this we attempt to find one of the intercepts with the coordinate planes (one of , , or being zero). Set to be zero, from the plane equation for , this tells us that . Then using the plean equation for , we get that . Therefore, the point is on the line. This tells us that we can write the equation of the line as
which is the parametric equation of the line.
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