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Don't have my book or audio lectures? You can download a free sample of my book and audio lectures containing Lessons 1, 2 and 9: Did you read my tips on how to study and learn Math
1300? If not, here is a link to those important suggestions: Here is a link to the actual assignment, in case you don't have it handy: Study Lesson 10 (Lines and Planes) from my Linear Algebra & Vector Geometry book to prepare for this assignment.
Please let me know if this assignment is revised. I am suspicious by the current version because it seems suspiciously short (only four questions and 30 marks).
Similar to my Lesson 10, question 4(b).
I am not sure if that is a typo where they ask for the "equations" of the plane. But, perhaps, they want you to give both the standard equation and the
point-normal form of the plane, as illustrated in my lesson.
Similar to my Lesson 10, question 4(a).
Part (b)
Unfortunately, I do not discuss the symmetric equation of a line in my book (I am surprised they
mention it here). Like all equations for lines in 3-space, R^3, you need a point on the line p = (x0, y0, z0) and a vector parallel to the line, v = (a, b, c). Then the symmetric equations for the line are written:
Which is to say, subtract the coordinates of the point from (x,y,z) in the respective numerators, and divide by the parallel vector (a,b,c) in the respective denominators.
The symmetric equations are derived from the parametric equations for a line. Merely solve
each parametric equation for t, and you end up with three different equations for t, which become the three symmetric equations for the line. That is why the three parts are listed as equal to each other.
Part (c)
Your course notes give you a formula for finding the distance between a point and a line that can be used here. Note that they give you four different formulas that could be used, two use the
sine function and the other two use cross products. You can use whichever one you wish. The prudent student would just memorize one of the cross product formulas and use it whenever they need to get the distance between a point and a line. That is far simpler than using trig. You will note the example they do uses one of the cross product formulas.
Again, use the formula I gave you above for the distance between a point and a line. To find the distance between two lines, merely generate a point on one line (it should be easy for you to read off one point on a line from the given equation for the line), and find the
distance between that point and the other line.
First, note that point-parallel form of a line is just another name for the vector form of a line.
Part (a)
Similar to my Lesson 10, question
9. Note that I use row-reduction to solve my problem (as taught in Lesson 2 of my book), but, technically, you aren't supposed to know that method yet. You can solve the problem using the methods I teach in Lesson 1 of my book, too. See my Lesson 1, question 8 for a similar example. You will note that, by introducing a parameter t into the solution, you end up getting the parametric equations for
a line. Now be sure to express those parametric equations as vector form (point-parallel form) by reading off p and v,
Part (b)
The dihedral angle is simply the angle made by the two planes. It is also the angle between the two planes' normal vectors. Simply use the formula for the cosine of an angle between two vectors as memorized in Lesson 9.
You will then have to use a calculator in order to find the actual angle (it is not a pretty answer). Make sure your calculator is in degree mode, then press inverse cosine (usually "2nd F" COS or "SHIFT" COS) to get the actual angle. I would round the answer off to one or two decimal places.
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