Fabian Waleffe -- Math 321 -- U. of Wisconsin

Math 321 homework, lecture pointers and learning objectives: SPRING 2009

CHAPTER 1: Vectors.

Numbered exercises refer to the Lecture notes on Vectors and Matrices that you have been studying. There are no problems to hand in. You should work on the problems listed below in preparation for exams.

Homework 1, Wed. 1/21/09 : STUDY SECTIONS 1.1, 1.2 focusing on geometric aspects
- Visualize/sketch the 8 properties (1)--(8) of vector addition and multiplication by a real number.
- What is a scalar ? Can a scalar be a function of time and/or position, or is it always a constant?
- What is Gibbs notation? How do we indicate magnitude of a vector? direction of a vector?
- Who is Gibbs? When and where did he live? What did he do in science and math?
- What are the units of a unit vector?
- What is the length of a unit vector? in what units of length?
Homework 2, Fri. 1/23/09 : STUDY SECTIONS 1.1, 1.2
- 1.2. 1, 5, 6, 7, 8. [5 and 6 done in class and sketched in the notes. Hide the solutions and try to do it from scratch on your own].
- Is a basis vector a unit vector?
- Is a unit vector a basis vector?
- Are basis vectors perpendicular to each other?
- What vectors are acceptable as a basis? dr/dt =
- For displacements in a plane, can "East" and "Northeast" serve as a basis? What about "2 miles East" and "3 miles Northeast"? If yes (for either or both cases), what are the components of "5 miles North" in that basis? [ANSWER]
- If you're OK with all this, go on and read sections 1.3 and 1.4. Always best to have an idea of what's coming next.
Homework 3, Mon. 1/26/09 : STUDY SECTIONS 1.3, 1.4
- Show/interpret all 5 properties of the dot product (page 7) from the geometric definition (11).
- 1.3. 1, 2*, 3, 4, 5, 6, 7*, 9*. In #5, perp means perp to some arbitrary n. Problems with a * are the `harder' problems.
- 1.4. Define δ_ij (Done in class 1/28)
- 1.4. Do you understand formula (14)? (discussed in class 1/28)
- 1.4. 1--5. (#4 done in class today with slightly different notation, #1, #2, #3, #5 (i), (ii) done in class W 1/28).
- Could e₁ be "East" and e₂ be "North"? What would e₃ be?
- Could e₁ be "Northeast" and e₂ be "Southeast"? What would e₃ be?
Homework 4, Wed. 1/28/09 : STUDY SECTIONS 1.3, 1.4, 1.5 (you *can* skip definitions of dot product and norms in Rⁿ on pages 9 and 10, although you will have to learn that sooner better than later in your education).
- What is the geometric definition of the vector product? What could that product be good for? What did you use that product for in physics or engineering courses? More info about right hand rule
- What is the difference between a . b and a x b ? Don't we use `.' or ` x ' to denote "product", as in 2 . 3 = 2 x 3 = 6?
- 1.5. 1, 2, 3, 6 (#2, #3 geometrically only for now, i.e. for 1/28/09).
Homework 5, Fri. 1/30/09 : STUDY section 1.5
- 1.5. 1, 2, 3, 4, 5, 6. #2, #3 geometrically and vector algebraically now (hint for #3: take a double cross).
- Could you show property 3 on page 11 from the geometric definition of the cross product?
- Can you indeed `obtain' (24) on p. 13?
- Sketch/explain geometrically what is (u x w) x u
- Is (u x w) x u = u x (w x u) ?
- Is (u x v) x w = u x (v x w) ?
- Points P₁, P₂, P₃ are specified in terms of their cartesian coordinates: (x₁,y₁,z₁), (x₂,y₂,z₂) and (x₃,y₃,z₃). Derive a formula for the area of the triangle AND for a direction normal to the triangle in terms of the cartesian coordinates. What is the formula for a vector A that is perpendicular to the triangle and has magnitude equal to the area of the triangle? (its `sense' is not defined, i.e. there are two answers to this question A and -A, either one is OK for this problem). dr/dt =
- *Consider any tetrahedron. It has 4 triangular faces, right? Define the 4 area vectors A_m, m =1, 2, 3, 4, such that A_m is perpendicular to face m pointing outward and has magnitude equal to the area of the triangle m. Show that A₁+A₂+A₃+A₄ =0. Visualize this in your head. Make a sketch.
- * What if you start packing up tetrahedra? e.g. Consider 5 points A, B, C, D, D'. They form tetrahedra A,B,C,D and A,B,C,D'. These are `glued' at the triangle ABC. So there are only 6 external faces and 6 area vectors A_m, m=1,...,6. Show that A₁+...+ A₆ = 0. Etc... Cool.
- The area of the triangle P₁, P₂, P₃ is given by the magnitude of 1/2 vec(P₁P₂) x vec(P₁P₃) (right?). If O is any point in the same plane, show geometrically and algebraically that the area is also given by the magnitude of 1/2 vec(OP₁) x vec(OP₂) + 1/2 vec(OP₂) x vec(OP₃) + 1/2 vec(OP₃) x vec(OP₁). Is that true also if O is NOT in the same plane?
- * Farmer Joe has laid out a fence. The poles are at cartesian locations (x_n,y_n), n=1,..., N. This is a closed pen so there is a straight barb wire from (x_n,y_n) to (x_n+1,y_n+1) and of course (x_N+1,y_N+1) ≡ (x₁,y₁) since the pen is closed, right?. Derive a nice compact formula for the area of the pen, Farmer Joe doesn't have time to think, he just wants to plug his GPS cartesian data into a formula.
Homework 6, Mon. 2/1/09 : STUDY section 1.6
- Read every word and re-derive every formula on page 14 and 15.
- How many equations does (28) stand for? What about (29)?
- Should you digest and memorize formulas (28) and/or (29)? Does one follow from the other?
- Expand out formula (31) and (32) into all their gory details.
- (33) and (34)?! What's going on here? Am I supposed to memorize these formulas?!!
- What is the definition of ε_ijk?
- A cyclic permutation of (a,b,c) is obtained by rotating the letters around (left or right) as if on a conveyor belt, i.e. (a,b,c) → (c,a,b) → (b,c,a) are the cyclic permutations of (a,b,c). Show that cyclic permutations of 3 elements ARE EVEN permutations of the elements. Show that cyclic permutations of 3 elements gives all even permutations.
- What are the cyclic permutations of 4 elements (a,b,c,d)? Show that cyclic permutations of 4 elements are ODD permutations. Show that cyclic permutations of 4 elements does NOT give all ODD permutations. How many odd permutations are there?
- The moral of the story is that "cyclic"= "even" in 3D, but not in higher dimensions.
Homework 7, Wed. 2/4/09 : STUDY section 1.6.2: Einstein's summation convention (all the following exercises where solved or at least sketched in class)
- Explain the summation convention to your roommates who did not take 321.
- Define the summation convention (s.c.)
- For a RHOB, expand (write) the vectors a, b, c using the s.c. dr/dt =
- Write a . b and a x b using the s.c., then expand out the index expressions, writing out all the sums explicitly.
- Write (a x b)_m using the s.c.
- Write (a x b). e_m using the s.c.
- Write (a x b) . c using the s.c.
- Write (a x b) x c using the s.c.
- Prove that a x b = - b x a using the s.c.
- Calculate ε_ijkε_ijk (s.c. assumed here and after)
- Calculate ε_ijkε_ijl
- Calculate ε_ijkδ_ij
- Calculate ε_ijkδ_kl
Homework 8, Fri. 2/6/09 : STUDY section 1.6: index notation and Einstein's summation convention
- Explain the summation convention to your roommates who did not take 321.
- You know an explicit formula for (a x b) x c valid for any vectors a, b, c (don't you?) Use that to deduce ε_ijkε_ilm = δ_jlδ_km - δ_jmδ_kl You should be able to reconstruct this formula, for different arrangements of indices. Can you deduce that formula ab initio ? or at least convince yourself it's right, using only the definition of ε_ijk and of the summation convention. I expect you to do that ( ab initio ) for ε_ijkε_ijk (3 dummies, 1 result) and for ε_ijkε_ijl (2 dummies, 9 results) but that's a bit less straightforward for me for ε_ijkε_ilm (1 dummy, 81 results). (Although it's clear that the result will be 0 if j=k or l=m. Then it's also pretty clear that the result will be 0 if the pair (j,k) is not a permutation of (l,m). Then it's clear the result is +1 only if (j,k)=(l,m) with j ≠ k and it's -1 only if (j,k)=(m,l) with j ≠ k. Hey! that's it, we took care of all the possible cases! I get it! not only can I reconstruct this formula easily, I also see why it is the way it is in two different ways).
- ε_ijkε_min = ? (that's (m,i,n) not "min"...)
- 1.6. 1, 2, 3, (all done in class multiple times), 4
- Write (a x b) . (c x d) in index notation, simplify/calculate out your index expression as much as possible, write your final expression in Gibbs/vector notation. Stop and enjoy. Or try to get the same result by computing cross products the way you did before Math 321. dr/dt =
- Which is correct, wrong, `kind of correct but not quite'? (i) v= v_j, (ii) v.e_j= v_j, (iii) v= v_j e_j, (iv) v= v_i e_j ?
Homework 9, Mon. 2/9/09 : STUDY section 1.7: Mixed products and determinants
- Is (a x b) . c a vector, a scalar or something else?
- What is the geometric meaning of (a x b) . c? Does the sign matter?
- What are the main properties of the mixed product?
- What is (a x b) . c in index notation (assuming that we're using a RHOB)?
- 1.7. 1, 2, 3, 7, 8, 9, 10, 11, 12, 13.
  Try to understand 15: what's going on geometrically, what's going on in terms of the algebra? This is (essentially) Gram-Schmidt, which you should learn about in linear algebra. (14 and 16 are basically the same problem, although there are various ways to solve them and get the same result. If you ever wonder how to measure areas and volumes in higher dimensional spaces, Gram-Schmidt and problems 10 and 16 give you a hint. In 321 we are almost always in dimension 3).
- Expand out (a x b) . (c x d) and (a + b) . (c + d). Do these obey the same algebraic rules? Don't mix them up! What about (a . b) x (c . d), what do you have to say about that?
- Explain algebraically and geometrically why det(a,b,c)= det(a, b+ βa , c+γa) for any β, γ
- Explain algebraically and geometrically why det(a,b,c)= det(a,b_⊥,c_⊥). "⊥" to what?!
- Suppose det(a,b,c)= D is known (D=-7, for instance). Then what is det(3 a + 2 b, b + 4c ,c + 5a )?
  What about det(α a + β b, b + γ c , c + ε a )?
- Is det(α a,αb,αc)= α det(a, b, c)? (α is a scalar). Explain algebraically and geometrically (Good question Wes!)
- Is α det(a, b, c) = det(α a,b,c)= det( a,α b,c)= det( a,b,αc) ? Explain algebraically and geometrically (Good question Wes!)
Homework 10, Wed. 2/11/09 : REVIEW ALL the sections that you should have STUDIED EARLIER: 1.1 to 1.7 included for exam 1 tomorrow Thursday Feb. 12, 5:00-6:15. But life goes on...
- Must know -- at least -- as well as how to spell your name:
  vector equations of points, lines and planes, as well as their translation into cartesian coordinates
- What is r? what is it's expression in cartesian coordinates? in index notation?
- What are |r| and r/|r| in cartesian coordinates?
- What are the geometric properties (e.g. "it is a sphere of radius 5 centered at the point (2,3,9)") of the set of points (x,y,z) that satisfy
  1. A₁ (x-x₁) = B₁ (y-y₁) = C₁ (z-z₁)? [Hint: no, it's not a sphere!]
  2. A₁ (x-x₁) + B₁ (y-y₁) = C₁ (z-z₁)? [Hint: no, it's not a sphere either!]
  3. (x-x₁)² + (y-y₁)² + (z-z₁)² = 0? [Hint: YES!! ... huh?!...]
- WAIT A MINUTE?!...
  THIS IS MATH 222 MATERIAL !!! You should know LINES AND PLANES IN VECTOR AND CARTESIAN FORMS!!
  What grade did you get in Math 222 anyway?
Homework 11, Fri. 2/13/09 : STUDY section 1.8 and your own lecture notes on those topics
1. equations of lines and planes in vector form and in cartesian coordinates form.
  WAIT A MINUTE?!...
  THIS IS ALL MATH 222 MATERIAL !!! You should know LINES AND PLANES!!
2. What is the distance between the point (x₁,y₁,z₁) and the plane a x + b y + c z + d = 0? (NOTE: If you simply look up a formula, you are cheating yourself. You should be able to handle this problem from scratch with no need to look up a formula. In other words, you should be able to derive the formula from scratch, be able to explain it and know how to use it.)
3. What is the distance between the point (4,2,9) and the plane 3x + 2y + z + 3 = 0? (See NOTE above)
4. How far is the plane a x + b y + c z + d = 0 from the origin? (See NOTE above)
  HAVING TROUBLE WITH THESE?! That means you don't REALLY understand equation of a plane!! Go back to HWK 10!
5. Answer this question, how about this one ? First question discussed in class Mon. 2/16/2009 . Here's the 2nd one:
6. 5% SUB: Quoting "InteractiveMind": ``I have a Light source located at point L and a `vertex' V. I then have a ray originating at the light source, and passing through the vertex. Therefore, the ray is defined as: r = L + t (V - L). I now need to find at what point this ray hits a plane...However, I'm not sure how to do this. My first problem is how I should define the plane, and then secondly, is how to actually locate the point of collision..''
  Formulate the problem and solve it. [`Obviously', the "solution" is not for one particular L and V and plane. We'd like to figure out the formula or, more generally, the algorithm needed to solve the problem for any given L, V and a plane].
7. ALL Problems on page 22 of the notes. Last problem discussed and solved on Mon. 2/16/2009
8. Rotation (rate) about an axis: made lots of pictures in class. You must understand the meaning of v = ω; x (r-r_A). Here v is velocity, ω; is the rotation vector, r is the position vector of the rotating particle, and r_A is the position vector of any reference point on the axis. Why are all those vectors linked in that way? can you explain that formula, not just regurgitate it? (if you understand it, you'll "know" it forever!)
9. 5% SUB: Finite rotations: point P has cartesian coordinates (3,2,1). What are its new coordinates after we rotate it by angle π/3 about the axis passing through point A whose coordinates are (1,2,3) and direction (4,5,1)? for the 5% SUB: Write the procedure as an "algorithm" (i.e. a `recipe' to follow to obtain the answer from the given data). Write a code to do this for any P, A, D, angle.
  [ANSWER for this data: new P has coords (0.9494, 3.6934, 0.7351)]
  We solved this easily (OK, not so easy perhaps, but not that hard thanks to cross product). You are responsible for understanding the procedure, you should be able to explain it to one of your peers, again the goal is NOT just to memorize a formula. This problem can be solved in a more complicated way using matrices. If you know linear algebra, stop and enjoy again the ease of the vector approach. Much simpler than the multiple matrix transformation approach of linear algebra.
Homework 12, Mon. 2/16/09 : STUDY section 1.9: vector function of a scalar variable, e.g. r=r(t) or any a=a(t).
NOTE some browsers do not display ω; properly this should be a bold "omega" meaning vector omega.
1. Can you prove a product rule? any of them on page 23?
  Look back in your Math 221 book to see WHY product rule is true. If you're going to use formulas, you want to convince yourself they are right!
2. Prove the derivative of a determinant rule using index notation.
3. What does a . da/dt=0 tell you about a? (i) if this eqn. applies at one specific time, (ii) if it applies for all times.
4. What does a x da/dt=0 tell you about a?
5. Suppose you know a and da/dt. Derive expressions for d|a|/dt and d/dt (a/|a|) in terms of a and da/dt (recall that (a/|a| is what we call and write a-hat). Do your equations say what you expect them to say? (i.e. interpret your results)
6. What is (d/dt) (r x m dr/dt), when m is constant?
  If r(t) is the position vector of a particle of mass m at time t, what is that equation about?
7. If v is the velocity of a particle at position r such that v=ω x r, what is the acceleration of the particle? What is the acceleration when ω is a constant? Do you know what those accelerations are called?
8. Consider the vector differential equation da/dt = ω x a, where ω is a constant vector, show that |a| and (ω . a) are constants. What is that telling you about a?
9. Solve the vector differential equation dr/dt = ω x r, where ω is a constant vector. What is that equation saying? Describe the motion in (a few) words. Is your solution saying what you think it should be saying?
10. Write dr/dt = ω x r in index notation and in cartesian coordinates.
11. Solve the vector differential equation dr/dt = ω x (r-r_A), where ω and r_A are (arbitrary) constant vectors. What is that equation saying? Describe the motion in (a few) words.
12. Write dr/dt = ω x (r-r_A) in cartesian coordinates.
13. ** Solve the vector differential equation dr/dt = ω x r + c, where ω and c are (arbitrary) constant vectors.
  Hint: write c = a ω - ω x r_A, what are a and r_A? why does that help you? Can you do this for any constant c?
14. If da/dt = ω x a, db/dt = ω x b and dc/dt = ω x c where ω is not necessarily constant, i.e. ω=ω(t), what is d/dt [det(a, b, c)] = ? Interpret geometrically, what's going on in this problem?
  The result is geometrically obvious and you should be able to see it without any calculations. The algebraic verification is a bit more involved but not too bad at all, if you know your mixed and double cross product identities. Hint: the Jacobi identity, problem 4, section 1.5 comes in handy.
Homework 13, Wed. 2/18/09 : Section 1.10 CATCH-UP!! Melody Pierson and Yaroslav Vergun solved `Interactive Mind''s problem (HWK 11). Svyatoslav Vergun solved the finite rotation problem (confirming that P' is (0.9494, 3.6934, 0.7351), Yeepie!!). Review and digest these, try to separate the concepts from the arithmetic (Svyat did a pretty good job of that having all the concepts on one side, then all the arithmetic on the other). All of those rotations and solutions to dr/dt = ω x (r-r_A) (i.e. rotation of a particle about an axis through A, with rotation vector ω) as well as: m dv/dt = q v x B (i.e. motion of an electric charge of charge "q" and mass "m" in a magnetic field B) are solved and discussed in these notes posted on top page .
Homework 14, Fri. 2/20/09 : CATCH-UP!! Today we found out that:
→ Ben COX, Noah ELMHORST, Ryan GRISA, Brent KARLEN, Will RAIFORD, and John WIETING skipped class !!!!
Hopefully they made good use of their time.
The rest of us learned how to code the problem of rotation about an arbitrary axis in MATLAB that Svyatoslav brilliantly solved for us on Wednesday (we clapped for him again!). See also OCTAVE developed here at UW in Chemical engineering, for a free alternative to MATLAB. The point was that the Matlab code is almost identical to our vector solution. We want to learn to `abstract' the key ideas and steps toward a solution, not spend our time computing dot and cross products `by hand' with a specific set of numbers.
We'll talk about Kepler and Newton on Monday. You may want to look at the history of mathematics before the `renaissance' started by Kepler, Fermat, Newton etc. That renaissance was indeed triggered by observations of planets by Galileo and Tycho Brahe in particular. Western mathematics, initated by the ancient Greeks, stopped abruptly with Hypathia around 400AD. Who was Hypathia? what happened to her? Yep, it's tough being a woman teaching mathematics!
Who knows what the story really is? History tends to be re-written. Who knows if Pythagoras was actually the one who found that theorem? But we certainly can believe Pythagoras' theorem and all the other Greek mathematics since we can check it and understand it for ourselves! Here's a cool proof of Pythagoras' theorem that makes uses of a key property of cross products: a x b = a x (b + α a ) for any real α. Do you see what that property is geometrically?
All of this of course is for Wes Hill, who's taking Math 321 but wants to major in "Ancient philosophy of Religion".
OUCH! some browsers do not display ω and ω properly. The first one is a bold "omega" meaning vector omega, the 2nd one is a lower case ω, it is the magnitude of the first. These appear in the following problem:
1. OK, after all my efforts, you still don't see that the solution of da/dt = ωn x a, where ω is a scalar constant and n is a constant unit vector (that is ωn is the rotation vector), is a(t) = [n . a(0)] n + cos (ω t) a_⊥(0) + sin (ω t) [ n x a_⊥(0)] where a(0) is the vector a(t) at time t=0 and a_⊥(0) is the component of a(0), perpendicular to ω, and you don't want to study the solution :-((
  but can you at least check it? HOW can you check it? [Hi Justin! did you read this far yet?]
2. We tried dividing a number by a vector and MATLAB did not like that at all! e.g. 1/[1,2,3] Try it.
3. We tried dividing a vector by a vector and MATLAB spat back a number, e.g. [1,2,3]/[4,5,1] =0.4048, what did Matlab do you think?! [Hint: it basically solved problem 1.3.9 without telling us! Wooooh! That's tricky! and maybe kind of dangerous, might make me think that dividing vectors by vectors is OK, while Matlab in fact made a non-trivial decision about what it thought I meant. How did it know that's what I meant? Why does it assume every Matlab user knows what s/he's doing!?!? Is that a safe assumption?]
4. but when we did [1,2,3]\[4,5,1] it gave us a matrix! What's going on?!?
Homework 15, Mon. 2/23/09 : Section 1.10: Kepler, Newton,... Wow! lots of concepts! Central force, cross product, dot product to get to constants of motions: angular momentum and `radius vector sweeping areas', Kinetic and potential energy...How does this all follow from Newton's law anyway? could you explain it to your peers?
Here are some nice illustrations of Kepler's laws, and the classical derivations from Newton's laws using polar coordinates. Hey! wait a minute, how do they even know that the motion is planar?! Shouldn't they start with spherical or cylindrical coordinates at least?
- Dig deeper: the 2-body problem , (basically 321 level)
- Dig even deeper: the 3-body problem and chaos! (Beyond 321 level, but not by that much, especially if you're also taking Physics 311) or take a look at
- the peer-reviewed 3-body problem sholarpedia page that has nifty animations (Beyond 321 level, but hey! this is a really important problem in the history of physics and mathematics. After you figure out motion of a single planet around the sun you'd of course worry about Sun + tiny Earth + big Jupiter, or Sun + Earth + small moon, i.e. a 3-body problem. Turns out to be much, much, much more complicated than just two planets! And are we sure the Earth won't go flying off into deep space one day, or be gobbled up by Jupiter?!)
- See equations of motions , find the equation for dr/dt. In what way is that that related to HWK 12 #5? (Remember since a vector has a magnitude and a direction, its rate of change has two contributions that follow from the product rule: one due to the change in magnitude, one to the change in direction. The latter is a rotation.).
ONWARD TO CHAPTER 2 WE GO!