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Math Help - Vectors and point on the line segment.. (3d)

  1. #1
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    Vectors and point on the line segment.. (3d)

    Two points A and B have position vectors (4i, -1j, 2k) and (1i, 5j, 3k). C is the point on the line segment AB such that AC / CB = 2. Find:

    b) the displacement vector AC.

    Please help me finding this displacement vector. I do know that it should be c - a or OC - OA but how would the rule fit in here.
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  2. #2
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    For AC/CB to be 2, C has to be placed at 2/3 of the line segment AB. The parametric equation for the line passing through AB is:

    (x,y,z)=A+tAB where t is the parameter. This gives the equation:

    (x,y,z)=(4,-1,2)+t(-3,6,1)

    At t=0, we get A and at t=1, we get B. Therefore, by setting t=2/3, we'll get our point C:

    C=(4,-1,2)+(2/3)(-3,6,1)

    Then, AC is given by:

    AC=C-A=(4,-1,2)+(2/3)(-3,6,1)-(4,-1,2)=(2/3)(-3,6,1)

    AC=(-2,4,2/3)
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  3. #3
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    For AC/CB to be 2, C has to be placed at 2/3 of the line segment AB. The parametric equation for the line passing through AB is:
    I don't understand why is it 2/3. How do I get it? I get it by solving algebraically but I need to understand how did you get it without using algebra at all.

    Thanks
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  4. #4
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    Quote Originally Posted by struck View Post
    I don't understand why is it 2/3. How do I get it? I get it by solving algebraically but I need to understand how did you get it without using algebra at all.

    Thanks
    The ratio of the line segments is 2:1. This means that it is split into thirds (as 2+1 = 3), with 2 of the thirds in the first segment (AC) and 1 of the thirds in the second segment (CB). Therefore, the placement of C must be \frac{2}{3} of the line segment AB.
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