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Math Help - Algebraic Vectors

  1. #1
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    Algebraic Vectors

    ABCD is a parallelogram with vertices A=(-1,2,-1), B=(2,-1,3), and D=(-3,1-3), find the coordinates of C.

    So this is my work below, BUT i cannot seem to get the correct coordinates of C, as my diagram does not seem to look exactly like a parallelogram..

    I sketched A, B, D..
    A is on the y-axis , D is below it to the right and in 4th quadrant, B is in second quadrant.. so I assumed that C would be in 2nd qadrant.

    The vector AC= The vector DB
    since they are parallel

    (x+1,y-2,z+1)=(5,-2,6)
    x+1=5
    x=4
    y-2=-2
    y=0
    z+1=6
    z=5

    C(4,0,5)
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  2. #2
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    First of all, be careful of the order of things.
    We want \overrightarrow {AD}  = \overrightarrow {BC} \;\& \,\overrightarrow {AB}  = \overrightarrow {DC} . Note the order.
    \overrightarrow {AD}  = \left\langle { - 2, - 1, - 2} \right\rangle  = \overrightarrow {BC}  = \left\langle {x - 2,y + 1,z - 3} \right\rangle
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  3. #3
    Member eXist's Avatar
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    Plato is correct. The vector AC is not parallel with the vector DB. Look at this picture (this is as if it were flattened into R^2):

    What I would do is find the equation of the line BC and the line DC, and then find their intersection. That intersection should be C.
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  4. #4
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    Quote Originally Posted by eXist View Post
    What I would do is find the equation of the line BC and the line DC, and then find their intersection. That intersection should be C.
    Actually all one has to do is to solve: \left\langle { - 2, - 1, - 2} \right\rangle = \left\langle {x - 2,y + 1,z - 3} \right\rangle .
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  5. #5
    Member eXist's Avatar
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    That's probably a better idea thanks.
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  6. #6
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    I'm having trouble drawing the parallelogram.. actually plotting the point A, B, and D into 3-space

    my parallelogram does not appear to be the same as yours, eXist, and AD and BC is not parallel in my drawing
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  7. #7
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    Quote Originally Posted by skeske1234 View Post
    I'm having trouble drawing the parallelogram.. actually plotting the point A, B, and D into 3-space
    my parallelogram does not appear to be the same as yours, eXist, and AD and BC is not parallel in my drawing
    There is really no good way to make this drawing.
    But if \overrightarrow {AD} \;\& \,\overrightarrow {BC} are not parallel in your drawing, then it is wrong anyway.
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  8. #8
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    well actually I am having trouble plotting the points like A(-1,2,-1) B(2,-1,3) and D(-3,1,-3) in 3 space, my teacher said to plot the vertex of the point first, and then you should have 3 lines forming a corner of a cube.. and then draw your cube, so that the arrow goes from orgin to opposite side of cube. is there a better method of doing this? because I seem to mess up the points where I am plotting A, B, and D, so then my parallelogram vertices are in the wrong quadrant .. thus wrong order
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  9. #9
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    Quote Originally Posted by skeske1234 View Post
    well actually I am having trouble plotting the points like A(-1,2,-1) B(2,-1,3) and D(-3,1,-3) in 3 space, my teacher said to plot the vertex of the point first, and then you should have 3 lines forming a corner of a cube.. and then draw your cube, so that the arrow goes from orgin to opposite side of cube. is there a better method of doing this? because I seem to mess up the points where I am plotting A, B, and D, so then my parallelogram vertices are in the wrong quadrant .. thus wrong order
    Well that makes no sense to me.
    There must be some good graphic program that would do that.
    But even the CAS I use does it poorly.

    But know this. Any three non-collinear points determine a triangle.
    Any triangle is ‘one-half’ of a parallelogram.
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  10. #10
    Member eXist's Avatar
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    Also look at the way ABCD is defined. The ordering of the letters should determine how the parallelogram is formed. With A touching B and D, B touching A and C, C touching B and D, and D touch C and A.
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