Vectors

Determine the vector equation of a line passing through the point P(3,2,-1) and with a direction vector perpendicular to the line r=(2,-3,4)+s(1,1,-2), seR

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  1. would the answer be r-(3,2,-1)+s(1,1,1)

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  2. Let's check:
    r1=(3,2,-1)+s<1,1,1>
    When s=0, r passes through the point (3,2,-1).
    Now check the product of the two direction vectors:
    <1,1,1> dot <1,1,-2>
    =1+1-2
    =0
    So the two lines are perpendicular
    and r1 satisfies the required conditions.

    Note: multiple lines can be perpendicular r, for example:
    r2=(3,2,-1)+t<5,3,4>
    would also work, since
    <5,3,4>dot<1,1,-2>=0 as well.

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