Science:Math Exam Resources/Courses/MATH152/April 2010/Question B 06 (c)

MATH152 April 2010

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Question B 06 (c)
Consider the triangle T in three dimensions with vertices (0, 1, 2), (1, 1, 5) and (−1, 2, 2). Consider also the plane P that contains T. Write an equation for P in the form $x+by+cz=d$ with b, c and d to be determined.

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

There are three unknowns and we know 3 vectors on P, so plugging them in will give a 3 by 3 linear system that can be solved by Gaussian elimination. However, this is not an efficient method. We know from part (b) that vectors lying in the plane P (meaning we can draw the heads and tails both in P) should all be perpendicular to the normal vector. Use this information to define an equation using dot product.

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. First pick a known vector on P. A convenient choice will be (0, 1, 2) (other vertices will work too). Let (x, y, z) be an arbitrary vector on P (this represents all possible vectors on P, drawn with tail at origin and head on P), then the difference (x, y, z) - (0, 1, 2) gives a vector lying in P. This vector must then be perpendicular to the normal vector, which gives the equation: $[(x,y,z)-(0,1,2)]\cdot (-3,-3,1)=0$ So $-3x-3y+z=(0,1,2)\cdot (-3,-3,1)=-1$ Dividing both sides by $-3$ gives $b=1,c=-{\frac {1}{3}},d={\frac {1}{3}}$