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You are not logged in. #1 20101230 03:21:38
Vector EquationsHi, And Respectively. The points A and B are transformed by the linear transformation T to the points A' and B' respectively. The transformation T is represented by the matrix T, where . a) Find the position vectors of A' and B'. I was able to work out that: . And . b) Hence find a vector equation of the line A'B'. Here is where I'm completely stuck . I haven't done any of the vector units which precede this one. #2 20101230 03:43:13
Re: Vector EquationsHi Au101; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #3 20101230 06:53:52
Re: Vector Equationshi Au101, lambda is the independant variable and is a scalar number; A'B' is the direction; OA' is the fixed point (you could use OB'); and r is the dependant variable. So find A'B' and substitute in the general vector equation. To get to any point on the line, first go from the origin to A'(OA') , and then take a variable trip along the vector A'B' (scalar.A'B') To test if your equation works, find the lambda that makes r = OB, then find the lambda that finds the midpoint of A'B' Bob You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #5 20101230 07:35:29
Re: Vector Equationshi Au101,
Bob Last edited by bob bundy (20101230 07:36:38) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #7 20101230 08:00:50
Re: Vector EquationsHIi Au101; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #9 20101230 08:31:18
Re: Vector Equationshi Au101 That's OB' followed by a 'different' lambda Or even: These are all the same line. Makes it hell to mark when every student may submit a correct, but different, equation. For this reason you may also find your answer looks different from the book answer. That's why it is worth trying to generate particular points on the line just to check it works. If you get one answer and the book another, they are the same when they generate the same points. Bob Last edited by bob bundy (20101230 08:32:40) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #10 20101230 08:32:58
Re: Vector EquationsSometimes it is a lot easier to learn by watching some videos rather than going through a book. They supplememnt each other well. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #11 20101230 23:18:49
Re: Vector EquationsHi Au101 and bobbym Last edited by bob bundy (20101231 00:33:26) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #12 20101231 00:04:19
Re: Vector EquationsHi Bob; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #13 20101231 00:30:46
Re: Vector Equationshi bobbym, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #14 20101231 01:29:07
Re: Vector EquationsThat's very nice . I think I get what the equation means now, thanks!!! . I am, however, stumped by a new, but similar form of question, I have a worked example, but as I've alluded to this is the last chapter of a unit which is way ahead of anything I've done and I don't really understand some of the ideas, perhaps someone could explain what to do and why? Is represented by the matrix T, where . The plane is transformed by T to the plane . The plane Has Cartesian equation Find a Cartesian equation of . Last edited by Au101 (20101231 01:29:41) #15 20101231 01:40:46
Re: Vector EquationsHi Au101, is perpendicular to the plane. (Do you need that explained because it is another post in itself). edit: I think what I said here was wrong. See post 20 for correction. Bob Last edited by bob bundy (20101231 05:16:20) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #16 20101231 01:48:02
Re: Vector EquationsHmmm such an explanation may indeed be helpful  the reason why I've moved so far in so short a time is that my textbook assumes knowledge of chapters I haven't yet covered. Also, I'm sorry to hear that you're out of bandwidth but I would be happy to have a look, I don't quite know where you mean by 'teaching resources' however? #17 20101231 01:51:03
Re: Vector Equationshttp://www.mathisfunforum.com/viewtopic.php?id=14858 Last edited by bob bundy (20101231 01:54:30) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #20 20101231 02:48:48
Re: Vector Equationshi Au101, Now suppose that vector 'n' is perpendicular to the plane. That means it is prependicular to every vector in the plane. Then But a.n = b.n = 0 as these vectors are perpendicular. So r.n = c.n = some constant as c and n are fixed. So if x,y and z are the components of r and n1, n2 and n3 are the components of n x.n1 + y.n2 + z.n3 = constant. This can be used to define the plane eg 3x + 4y  6z = 27 (iii) Planes that go through (0,0) Matrix multiplication can only be used for transformations that leave the origin invariant ie. M . 0 = 0 So your transform must be one of these otherwise they couldn't set this question. And we can see that the origin is a point in the plane as x = y = z = 0 fits. So the plane constant must be zero So is a possible vector for n, perpendicular to the plane. (Or any multiple of n) Bob Last edited by bob bundy (20101231 06:20:17) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #21 20101231 06:14:27
Re: Vector Equationshi step (i) By inspection these vectors will both lie in the plane step (ii) Transform these to get step (iii) so the equation is: so A x 6 => Then add C also add B and C E x 8 => add D Bob Last edited by bob bundy (20101231 06:15:11) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #23 20101231 07:47:12
Re: Vector Equationshi Au101, So I wanted two vectors (chosing x, y and z) that would fit that equation. I just spotted x=y=z=1 as one possibility and then x = 1 y = 0 and z = 1 as another. I was pleased to choose those because the numbers were easy to work with. But any three numbers that fit the equation would do for a vector in the plane: Let's have x = 39 and y = 22. Then z = 44  39 = 5. That's what I meant before by two degrees of freedom. Choose any two and that fixes the third. So z = 3, y = 100 => x = 200 + 3 = 203. And so on. But 1,1,1 and 1,0,1 were easier to work with. The only choice you cannot make is to have two vectors that are parallel, eg. x = y = z = 1 for one choice and x = y = z = 2 for the other. They are valid choices but you cannot solve the problem because you don't get enough independent equations to solve for mu and lambda. Look back to my diagram for the plane. If vectors a and b are parallel you would not be able to get to all possible places for D. The vectors have to be 'independent' (meaning one cannot be made by multiplying the other by a fixed amount). I expect you'll meet that idea again in this module. Bob Last edited by bob bundy (20101231 07:54:27) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei 