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Math Help - simple explanation about translation vectors

  1. #1
    Senior Member furor celtica's Avatar
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    simple explanation about translation vectors

    1. If p = 2i + 3j, q = 4i – 5j and r = i – 4j, find a set of numbers f, g and h such that fp + gq + hr = 0. Illustrate your answer geometrically. Give a reason why there is more than one possible answer to this question.
    2. If p = 3i - j, q = 4i + 5j and r = -6i + 2j
    a. can you find the find numbers s and t such that q = sp + tr
    b. can you find numbers u and v such that r = up + vq?
    Give geometrical reasons for your answers.

    As you can see these are pretty basic vector problems. I didn’t have any difficulties with the strictly algebraic part of these questions, and my answers are as follows:
    1. f = 1, g = -1, h = 2 ( I am aware that any multiples of these numbers would do as well.)
    2. a. No.
    b. Yes: u=2, v=0

    However I’m not sure what they mean by, ‘giving reasons’. This sounds very vague to me, and I’d like to know what sort of answer I’m supposed to be giving, along what lines this ‘proof’ is supposed to be.
    Here is my attempted explanation for 2.a.:

    r is a multiple of p, which means that the result of sp + tr must be a multiple of p; q is not a multiple of p therefore numbers s and t do not exist for which q = sp + tr. (At this point I remembered I was supposed to give ‘geometrical reasons’ (what is that supposed to mean anyway?) so I added this)
    Geometrically, points (0i, 0j), (3i, -j) and (-6i, 2j) are collinear: sp, tr and q will not meet thrice and a triangle will not be formed.

    As you’ve noticed I’m not really sure where I’m going with this explanation (especially towards the geometrical part, that was just embarrassing to post), so if I had a few tips on what sort of ‘reasons’ I’m looking for in all three questions that would help a lot. Cheers
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  2. #2
    Grand Panjandrum
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    Re: simple explanation about translation vectors

    Quote Originally Posted by furor celtica View Post
    1. If p = 2i + 3j, q = 4i – 5j and r = i – 4j, find a set of numbers f, g and h such that fp + gq + hr = 0. Illustrate your answer geometrically. Give a reason why there is more than one possible answer to this question.
    2. If p = 3i - j, q = 4i + 5j and r = -6i + 2j
    a. can you find the find numbers s and t such that q = sp + tr
    b. can you find numbers u and v such that r = up + vq?
    Give geometrical reasons for your answers.

    As you can see these are pretty basic vector problems. I didn’t have any difficulties with the strictly algebraic part of these questions, and my answers are as follows:
    1. f = 1, g = -1, h = 2 ( I am aware that any multiples of these numbers would do as well.)
    2. a. No.
    b. Yes: u=2, v=0

    However I’m not sure what they mean by, ‘giving reasons’. This sounds very vague to me, and I’d like to know what sort of answer I’m supposed to be giving, along what lines this ‘proof’ is supposed to be.
    Here is my attempted explanation for 2.a.:

    r is a multiple of p, which means that the result of sp + tr must be a multiple of p; q is not a multiple of p therefore numbers s and t do not exist for which q = sp + tr. (At this point I remembered I was supposed to give ‘geometrical reasons’ (what is that supposed to mean anyway?) so I added this)
    Geometrically, points (0i, 0j), (3i, -j) and (-6i, 2j) are collinear: sp, tr and q will not meet thrice and a triangle will not be formed.

    As you’ve noticed I’m not really sure where I’m going with this explanation (especially towards the geometrical part, that was just embarrassing to post), so if I had a few tips on what sort of ‘reasons’ I’m looking for in all three questions that would help a lot. Cheers
    1. Because these vectors are pairwise linearly independent (please check that they are) they may be taken as the directions of the sides of a triangle, and that triangle may be of any size.

    To find a set of numbers for which this holds left f=1 and solve the resulting pair of equations to find g anf h.

    CB
    Last edited by CaptainBlack; January 10th 2012 at 10:41 PM.
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  3. #3
    Senior Member furor celtica's Avatar
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    Re: simple explanation about translation vectors

    what's linearly independent? i.e. not on the same line?
    Is the answer I gave acceptable?
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  4. #4
    Grand Panjandrum
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    Re: simple explanation about translation vectors

    Quote Originally Posted by furor celtica View Post
    what's linearly independent? i.e. not on the same line?
    Is the answer I gave acceptable?
    Pairwise linearly independednt - nn pair are parallel.

    Other than not providing a geometric explanation for 1 and any explanation for 2a, they are acceptable.

    CB
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  5. #5
    Senior Member furor celtica's Avatar
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    Re: simple explanation about translation vectors

    you mean no explanation for 2b? also is it correct to use the notation "points (0i, 0j), (3i, -j) and (-6i, 2j)" when dealing with translation vectors?
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  6. #6
    Grand Panjandrum
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    Re: simple explanation about translation vectors

    Quote Originally Posted by furor celtica View Post
    you mean no explanation for 2b? also is it correct to use the notation "points (0i, 0j), (3i, -j) and (-6i, 2j)" when dealing with translation vectors?
    2b does not need an explanation since you give the valuse that answer the question (assuming they are right)

    If you give a vector as an ordered pair there is no need for the i and j's, but you will be better off sticking to whatever notation your instructor or text book uses.

    CB
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