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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 2: Did you read my tips on how to study and learn Math 1300? If not, here is a link to those important suggestions: Did you see my tips for Assignment 1? 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.
Lesson 10 is a very challenging lesson and demands that a student is able to visualize the problems. I STRONGLY RECOMMEND that you consider spending the $20 to purchase my audio lectures for this course. I think you will find the audio for this lesson very helpful in your goal of truly understanding these topics.
Similar to my Lesson 10, question 3. Except they want the point-normal form and the standard form, so be sure to give them both as illustrated in my lesson.
Similar to my Lesson 10, question 4. The two-point vector form is just what I call vector form. Again, they want both vector form and parametric form for the line in (b), so give them both.
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.
Skew lines are lines that are not on the same plane (therefore they will never intersect), but are not parallel. For example, visualize a line drawn from east to west on the floor. Now visualize drawing a diagonal line on the ceiling. Those lines are not
parallel to each other, but they would never intersect either. Keep that visual image in your head as you do this problem.
The acute angle is just the angle the two lines would make if they did share the same plane (like if my second line on the ceiling had also been drawn on the floor diagonally, intersecting that east-west line on the floor). Otherwise, this is similar to question 3, but now use the two vectors you can read off the lines. Note, if
you have done it right, you will get a negative dot product which will lead to an obtuse angle (since cosine is negative when the angle is between 90 and 180 degrees). Merely subtract this angle from 180 to get the acute angle they request.
Visualize a horizontal line, and then another line intersecting it at a 45 degree angle, for example. Depending how you look at it, the angle between the two lines is either 45 degrees or 135 degrees (the
supplement). They prefer that you give them the acute angle.
Part (a)
Similar to my Lesson 10, question 10(a).
Part (b)
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. Part (a)
As I discuss in Lesson 9, the cross product of two vectors gives you a vector orthogonal to both of them. Use the vectors associated with each of the given lines.
Part
(b)
Just use the formula for the sine of an angle between two vectors as taught in Lesson 9. This also ties in nicely with the cross product you just did.
Part (c)
Similar to Assignment question 5(b). Merely choose any point you wish on the first line, and find the distance between that point and the second line (or vice-versa). If you think about it, that
is what the diagram in the link I gave you from your notes is trying to show you. You are almost ready to answer this question once you have done the previous two parts.
Part (a)
Similar to my Lesson 10, question 6.
Part (b) Similar to my Lesson 10, question
5.
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