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Math Help - coordinate system

  1. #1
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    Post coordinate system

    Hey All,

    How would you solve the following vector question:

    In a cartestian coordinate system, point A is given by (-3, 0, -5) and a point B by (3, 4, 3). Find the direction cosines of the line AB and write down the vector equation for this line.

    A second line passes through point C with coordinates (2, 5, 5), and is in the direction (2i + 3j + 6k). show that these two lines intersect at a point D and write down the coordinate of D.

    Find a vector that is perpendicular to both AB and CD.

    Thank you
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  2. #2
    Member Glaysher's Avatar
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    Quote Originally Posted by dadon View Post
    Hey All,

    How would you solve the following vector question:

    In a cartestian coordinate system, point A is given by (-3, 0, -5) and a point B by (3, 4, 3). Find the direction cosines of the line AB and write down the vector equation for this line.

    A second line passes through point C with coordinates (2, 5, 5), and is in the direction (2i + 3j + 6k). show that these two lines intersect at a point D and write down the coordinate of D.

    Find a vector that is perpendicular to both AB and CD.

    Thank you
    Vector equation of AB

    r = point on AB + t(direction of AB)

    r = (-3i - 5k) + t(6i + 4j + 8k)

    [Direction of AB = position vector of point B - position vector of point A]

    Second line has vector equation

    s = (2i + 5j + 5k) + u(2i + 3j + 6k)

    Make two vector equations equal to each other and equate coefficients

    (-3i - 5k) + t(6i + 4j + 8k) = (2i + 5j + 5k) + u(2i + 3j + 6k)

    giving

    for i

    -3 + 6t = 2 + 2u

    for j

    4t = 5 + 3u

    for k

    -5 + 8t = 5 + 6u

    Take two of these three equations and solve simultaneously to find t and u

    Stick the found values of t and u into the third equation

    If they work the lines intersect, if they don't the lines don't intersect

    Stick either your value of t into r = (-3i - 5k) + t(6i + 4j + 8k) or your value of u into s = (2i + 5j + 5k) + u(2i + 3j + 6k) to find D
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  3. #3
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    Quote Originally Posted by Glaysher View Post
    r = (-3i - 5k) + t(6i + 4j + 8k)
    Hi,

    How do you get (-3i -5k)

    are you using the dot or cross product rule?

    Thanks
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  4. #4
    Member Glaysher's Avatar
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    Quote Originally Posted by dadon View Post
    Hi,

    How do you get (-3i -5k)

    are you using the dot or cross product rule?

    Thanks
    Neither, (-3i -5k) reprsents position vector of point A (-3, 0, -5)
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  5. #5
    Member Glaysher's Avatar
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    Quote Originally Posted by dadon View Post
    Hey All,

    How would you solve the following vector question:

    In a cartestian coordinate system, point A is given by (-3, 0, -5) and a point B by (3, 4, 3). Find the direction cosines of the line AB and write down the vector equation for this line.

    A second line passes through point C with coordinates (2, 5, 5), and is in the direction (2i + 3j + 6k). show that these two lines intersect at a point D and write down the coordinate of D.

    Find a vector that is perpendicular to both AB and CD.

    Thank you
    Cross product of (6i + 4j + 8k) and (2i + 3j + 6k) gives vector perpendicular to AB and CD
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