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Math Help - Vector Calculations

  1. #1
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    Vector Calculations

    Hello all, not sure if this is the right thread but i need to check some of my answers to some problems. thanks.

    if u = i + 2j - k and v = j - 3k are vectors in R^3. Find:

    a) a unit vector poiting in the direction of -3u:

    my solution: -3 (i + 2j - k) = -3i - 6j +3k

    so then the magnitude is: root (9 +36 + 9) which is root (54) so then the unit vector is 1 / root(54) * (3i + 6j - 3k)

    b) u * (12v)

    i used the fact that u * (12v) is equal to 12 (u * v)
    i first found u * v using the dot product formula to get 5. so then 12 (5) = 70.

    c) the cosine of the angle between u and v.

    I used the product i found earlier (5) and used the formula cos theta = u * v / (absolute value of u * absolute value of v) and got the answer 5 / root (60)
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  2. #2
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    Re: Vector Calculations

    Quote Originally Posted by andrew2322 View Post
    Hello all, not sure if this is the right thread but i need to check some of my answers to some problems. thanks.

    if u = i + 2j - k and v = j - 3k are vectors in R^3. Find:

    a) a unit vector poiting in the direction of -3u:

    my solution: -3 (i + 2j - k) = -3i - 6j +3k

    so then the magnitude is: root (9 +36 + 9) which is root (54) so then the unit vector is 1 / root(54) * (3i + 6j - 3k)

    b) u * (12v)

    i used the fact that u * (12v) is equal to 12 (u * v)
    i first found u * v using the dot product formula to get 5. so then 12 (5) = 70.

    c) the cosine of the angle between u and v.

    I used the product i found earlier (5) and used the formula cos theta = u * v / (absolute value of u * absolute value of v) and got the answer 5 / root (60)
    a) is correct, but you can simplify it further.

    b) Is this a dot product or a cross product? Assuming it's a dot product (you should use a dot instead of an asterix), you are correct.

    c) You are correct that \displaystyle \cos{\theta} = \frac{5}{\sqrt{60}} which you can simplify more. What is \displaystyle \theta?
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    Re: Vector Calculations

    Hello Prove it.

    Yes, B is the dot product, i apologise for the ambiguity.

    and with c, if the question asks for the cosine of the angle, does that mean they want cos theta or just theta alone?
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    Re: Vector Calculations

    Quote Originally Posted by andrew2322 View Post
    Hello all, not sure if this is the right thread but i need to check some of my answers to some problems. thanks. if u = i + 2j - k and v = j - 3k are vectors in R^3. Find:
    a) a unit vector poiting in the direction of -3u:
    my solution: -3 (i + 2j - k) = -3i - 6j +3k
    so then the magnitude is: root (9 +36 + 9) which is root (54) so then the unit vector is 1 / root(54) * (3i + 6j - 3k)
    I actually think that your answer there is not be correct.
    Now mind you, it may be just the way I am reading the question.
    "a) a unit vector pointing in the direction of -3u"
    The word 'pointing in the direction as -3u means in the opposite direction as u. That is the effect of the negative.
    Positive multiples of u are in the same direction, negative multiples are in the opposite direction
    You answer above is a positive multiple.

    If u=i+2j-k then \frac{-1}{\sqrt6}u is a unit vector parallel to -3u in the same direction.
    That is all part a) is asking for. You did a lot of unnecessary steps.
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    Re: Vector Calculations

    I understand what you mean Plato, but is it possible that given the information we have, that both solutions are correct in their own right?
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    Re: Vector Calculations

    Sorry, i think there was a typo in my answer. It was meant to be [-1 / 3 * root(6)] * (3i + 6j - 3k)
    which is a unit vector pointing in the right direction as -u.

    is this correct?
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    Re: Vector Calculations

    Quote Originally Posted by andrew2322 View Post
    is it possible that given the information we have, that both solutions are correct in their own right?
    I don't know how to answer that? Consult your text material and/or instructor.

    I will say that these are typical questions.
    If the question were "write a unit vector in the direction of -3\vec{u}" then the simple and correct answer would be -\frac{\vec{u}}{\|\vec{u}\|}.
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    Re: Vector Calculations

    which is equivalent to - (1/3) * (1/ root 6) ( 3i + 6j - 3k) which is the same as (1 / root 6) (i + 2j - k)

    is the math correct?
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