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Math Help - Check my work regarding vector magnitude and direction?

  1. #1
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    Check my work regarding vector magnitude and direction?

    Sorry if this is the wrong section, but it came from my Calc book.

    The question is : find a vector of norm 10\sqrt{2} in the direction opposite of 3i + 4j - 5k.

    As I understand it, the direction vector is the \mathbf{V}/\left \| \mathbf{V} \right \|

    So the direction vector of the given vector is:

    3i/\sqrt{50} + 4j/\sqrt{50} - 5k/\sqrt{50} right?

    New vector needs to be opposite so switch the signs:

    -3i/\sqrt{50} - 4j/\sqrt{50} + 5k/\sqrt{50}

    Now do I multiply the required norm into this direction vector to get the final answer:?

    -30\sqrt{2}i/\sqrt{50} - 40\sqrt{2}j/\sqrt{50} + 50\sqrt{2}k/\sqrt{50}

    I just took a stab at it, so I likely messed up...
    Last edited by Mattpd; June 30th 2010 at 04:15 PM.
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  2. #2
    Senior Member jakncoke's Avatar
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    if u take the vector  \begin{bmatrix} 3 \\ 4 \\ -5 \end{bmatrix} and multiply it by -2. u get a vector  \begin{bmatrix} -6 \\ -8 \\ 10 \end{bmatrix} with a norm of 10*root(2) in the oppsite direction. The direction of the vector is simply 3 in the x direction, 4 in the y direction, -5 in the z direction. the norm is  \sqrt{x^2 + y^2 + z^2} . \mathbf{V}/\left \| \mathbf{V} \right \| is the formula for the unit vectors in the given direction, meaning its the same direction as the original vector but just scaled to have unit magnitude.
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  3. #3
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    Can I ask how you knew to multiply by -2?
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  4. #4
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    Jakncoke had measured the length of your original vector and found it to be 5\sqrt{2}. So you had to magnify it by 2 to get the length of 10\sqrt{2}, and then by -1 to reverse its direction.
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  5. #5
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    Quote Originally Posted by Mattpd View Post
    Sorry if this is the wrong section, but it came from my Calc book.

    The question is : find a vector of norm 10\sqrt{2} in the direction opposite of 3i + 4j - 5k.

    As I understand it, the direction vector is the \mathbf{V}/\left \| \mathbf{V} \right \|

    So the direction vector of the given vector is:

    3i/\sqrt{50} + 4j/\sqrt{50} - 5k/\sqrt{50} right?

    New vector needs to be opposite so switch the signs:

    -3i/\sqrt{50} - 4j/\sqrt{50} + 5k/\sqrt{50}

    Now do I multiply the required norm into this direction vector to get the final answer:?

    -30\sqrt{2}i/\sqrt{50} - 40\sqrt{2}j/\sqrt{50} + 50\sqrt{2}k/\sqrt{50}

    I just took a stab at it, so I likely messed up...
    It helps to know that 50= 2*25 so that \frac{\sqrt{2}}{\sqrt{50}}= \frac{\sqrt{2}}{\sqrt{2}\sqrt{25}} = \frac{1}{\sqrt{25}}= \frac{1}{5}
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