course Mth 275
Resubmission, no response received. Original submitted on 6/18 or 19.
Section 10.4:I understand the material in the section. I do have one question: is the cross product associative [e.g.: is (u X v) X w = u X (v X w)]? Intuitively, it seems that it should be, but I? not sure.
I apologize for the delay in my response. I had to put the question aside until I got a pen and paper, and as sometimes happens ended up emailing my partial response to myself rather than to you. I'll give you more of a response on the last question when I get back in town Friday (later tonight if I get caught up with my responses earlier than I expect).
see also email for access code etc
course Mth 275
Now that school has ended, I'll pick up the pace.
Section 10.4:I understand the material in the section. I do have one question: is the cross product associative [e.g.: is (u X v) X w = u X (v X w)]? Intuitively, it seems that it should be, but I’m not sure.
If u = u1 i + u2 j + u3 k, v = v1 i + v2 j + v3 k and w = w1 i + w2 j + w3 k, then u X v = (u2 v3 - u3 v2) i + (u3 v1 - u1 v3) j + (u1 v2 - u2 v1) k.
So (u X v) X w = [ (u2 v3 - u3 v2) i + (u3 v1 - u1 v3) j + (u1 v2 - u2 v1) k ] X (w1 i + w2 j + w3 k). You can write it all out if you're careful about the bookkeeping.
You can do the same for u X ( v X w).
This will yield insights into the patterns of the expressions and into cases where associativity will and will not be the case.
Alternatively you can consider a few combinations of three simpler vectors. For example if you let
u = i, v = (i + j), w = j
then since i X i = j X j = k X k = 0, i X j = k, i X k = -j and j X k = i you get
(u X v) = i X (i + j) = i X i + i X j = k and
(u X v) X j = k X j = -i but
since v X w = (i + j) X j = i X j + j X j = k we have
u X (v X w) = i X k = -j.
You can also visualize these vectors and the directions of the various cross products, which yields additional insight into why associativity does not in general hold.
Problem #42: “Find the angle between the vector 2i – j + k and the plane determined by the points P(1, -2, 3), Q(-1, 2, 3), and R (1, 2, -3).” I used the section on the volume of a parallepiped, specifically (u X v) dot w = ||u X v|| ||w|| cos theta. I computed u = PQ = -2i+4j and v = PR = 4j-6k. Then I found u X v = -24i - 12j - 8k, ||u X v|| = 28, ||w|| = sqrt 6, and (u X v) dot w = -44. Substituting in (u X v) dot w = ||u X v|| ||w|| cos theta, I found theta approx. = 130 degrees. Since theta is the angle between (u X v) and w, I subtracted 90 degrees, giving an angle of 40 degrees between the vector and the plane. Is this a correct solution?
This is the approach I would have suggested. Assuming you haven't made any arithmetic errors this will be correct.
I would describe the process as: Find a vector normal to the plane (cross product of PQ and PR), find the angle between the given vector and the normal using the definition of the dot product, and subtract 90 degrees.
I was unable to solve #63: “Establish the validity of u X [u X (u X v)] dot w = - ||u||^2u dot v X w.” I tried using “cab – bac” on u X (u X v), but I couldn’t get anywhere with the result of that calculation.
I'll have to get back to you on this one, simply because it's going to take awhile to type out a decent explanation. Briefly, u is perpendicular to anything u is crossed with, which leads fairly quickly to the |u|^2 part.