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Math Help - Vector equation!!!

  1. #1
    Junior Member
    Joined
    Mar 2008
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    56

    Vector equation!!!

    Hi everyone. I'm having some problems with this couple examples.

    Let OAB be a triangle, that is, O,A and B are not collinear. Now let R and S be the mid-points of the sides AB and OA respectively and let M be the point of intersection of the line segments OR and BS.

    a)
    Express the vector OS as a linear combination of OA and OB.

    Is it : OS = OB - 1/2(OA) ??

    b)
    Express the vector OR as a linear combination of OA and OB.

    Is it: OR = OB + 1/2(BA) = OB + 1/2(OA - OB) = 1/2(OA + OB)??

    c)

    Give the vector equation of the line through O and R in terms of OA and OB.

    Is it: OR = OB + BR = OB + tBA???

    d)
    Give the vector equation of the line through B and S in terms of OA and OB.

    e)
    Express the vector OM as a scalar multiple of OR.

    thanks for any help.

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  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Hi,

    Your a) and b) are correct. As for the following, the important is that the equation of a line through A and B may be written geometrically as: \overrightarrow{AP}=t\overrightarrow{AB}, t\in\mathbb{R} (meaning that the line is the set of all points P such that ... for some t\in\mathbb{R}). You may also write it P=A+t\overrightarrow{AB}, t\in\mathbb{R}, if you used this notation in your geometry lesson.

    So, for c), the line through O and R has the following equation: \overrightarrow{OP}=t\overrightarrow{OR}=\frac{t}{  2}(\overrightarrow{OA}+\overrightarrow{OB}), t\in\mathbb{R}.

    Try to take it from here. d) works the same, and in e) involves writing that M satisfies both previous equations (with a priori distinct parameters t and t'), and eliminating M by combining these equations, in order to find t and t'.

    Laurent.
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