Math 225: Calculus III     Assignment 2

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I affirm that the solutions presented in this assignment are entirely my own work.

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Instructions

This assignment contains problems on vectors, dot and cross products, and lines and planes in space, from Chapter 11 in Finney & Thomas and Chapter 1 in the Lecture Notes. You should look in AdvanceCalculusDemo and Exercises in the Notebook folder or the back of the Lecture Notes as well as the the early quiz notebooks in the Quizzes folder for examples of how to do the following problems. The functions Norm[], Cross[], and Pr[] are explained in AdvanceCalculusDemo. 

Problem 4 can be solved using cross products. You should use the Solve[] command in Problem 5.

Don't forget to Clear[] or Remove[] variables before moving on to the next problem.

Be sure to type in comments explaining what you are doing. It is best to change the style of such comments to Text. To do this, select the cell by clicking on the bar at the right and select Text from the pop-up menu in the ruler above (or hit COMMAND-7).

Remember to uncheck Show In/Out Names in the Kernel menu and close this group before printing.

Initialization

<<Calculus`Math225`

i = {1, 0, 0};
j = {0, 1, 0};
k = {0, 0, 1};

Problem 1

Let  a = i - 2j + 3k and  b = -i + 4j + k. Find the following.
a)   Norm[a], Norm[b]
b)  a.b,  b.a
c)  Cross[a,b], Cross[b,a]
d)  The angle between a and b, and the projection of b on a (Pr[b,a]).

Solution

Problem 2

a)  Find the area of the parallelogram determined by
     a = i + j - 3 k and b = -2 i + j + k
  using a cross product.
b)  Find the volume of the parallelopiped determined by a, b, and
     c = i - 2 j + k using a triple products (or a determinant, Det[]).

Solution

Problem 3

a)  Find the distance from the point p = {1,-3,2} to the plane
    5x + y - 4z == 1.
b)  Find the distance from the point q = {2,1,3} to the line
    x == 1 - t, y == 1 + t, z == 3t.

Solution

Problem 4

a)  Find an equation for the plane through the points p = {1,-1,2},
    q = {3,0,1}, and r = {-1,1,1}.
b)  Find the parametric equations for the line in which the planes x - y + z == 1
    and x + y + 3z == -2 intersect.

Solution

Problem 5

a)  Find the point in which the line x == 2-2t, y == 1-t, z == 3+5t meets the plane 3x + 2y + 6z == 4.
b)  Find the point in which the three planes x - y + z == 1,
x + y - z == 2 and -x + y + z == 3 intersect.

Solution