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Math Help - A brief vector question

  1. #1
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    A brief vector question

    This vector question has three parts, and I've done the first two, finding the point of intersection and acute angle between them. It's the third bit I don't understand.

    L1 = 2i-j+3k + s(j-k)
    L2 = -2i+7j+3k + t(2i-3j-k)

    iii) The foot of the perpendicular from the point with position vector -2i+7j+3k to the line L1 is P. Find the position vector of P.

    I just don't really understand what this means, any help would be much appreciated.
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  2. #2
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    Quote Originally Posted by JeWiSh View Post
    This vector question has three parts, and I've done the first two, finding the point of intersection and acute angle between them. It's the third bit I don't understand.

    L1 = 2i-j+3k + s(j-k)
    L2 = -2i+7j+3k + t(2i-3j-k)

    iii) The foot of the perpendicular from the point with position vector -2i+7j+3k to the line L1 is P. Find the position vector of P.

    I just don't really understand what this means, any help would be much appreciated.
    1. Let P denote the foot of the perpendicular line to L1 and Q(-2, 7, 3). Then PQ is the shortest distance between the line and Q.

    2. Calculate the distance between Q and any arbitrary point on L1:

    d(s) = \sqrt{\left((2, -1, 3) + s\cdot (0, 1, -1) - (-2, -7, 3)\right)^2} = \sqrt{\left((4, -8, 0)+s\cdot(0, 1, -1)\right)^2}

    3. Let D(s) = (d(s))^2. Calculate now the minimum of D:

    D(s)=2s^2 - 16s + 80~\implies~ D'(s)=4s-16

    Therefore s = 4

    4. Plug in this value into the equation of L1 to calculate the coordinates of P:

    \vec F  = 2i-j+3k + 4 \cdot (j-k) = (2i + 3j - 1k)
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  3. #3
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    This vector question has three parts, and I've done the first two,
    finding the point of intersection and acute angle between them.
    It's the third bit I don't understand.

    L_1\!:\;\;2i-j+3k + s(j-k)

    iii) The foot of the perpendicular from the point A
    with position vector: -2i+7j+3k to line L_1 is P.
    Find the position vector of P.
    Code:
               A
                *
                 \
                  \
                   \               L1
                    \           *
                     \     *
                      *
                 *     P
            *

    AP is drawn perpendicular to L_1

    We are asked to locate point P.

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