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Math Help - Points, angles and lines of intersection.

  1. #1
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    Points, angles and lines of intersection.

    a) Determine where the line L : x = (-1, 8, 9)+t(3,-9,-4) and the plane
    E : -2x1 + 6x2 + 3x3 + 19 = 0 intersect and calculate the angle between
    them.
    b) Given the two planes
    E : . = 2.x + 3x2 + 7x3 + 2 = 0, E : x=(3,0,0)+ s(-2,0,0)+t(3,0,1)
    find the line of intersection and the angle between them.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    What have you tried?

    Quote Originally Posted by mamt6 View Post
    a) Determine where the line L : x = (-1, 8, 9)+t(3,-9,-4) and the plane
    E : -2x1 + 6x2 + 3x3 + 19 = 0 intersect and calculate the angle between
    them.
    Note that we can write the line as, L = \left<-1 + 3t, 8 - 9t, 9 - 4t \right>.

    That is, x_1 = -1 + 3t,~~x_2 = 8-9t,~~x_3 = 9 - 4t

    You can use that to find the point of intersection. As for the angle of inetersection, use the formula

    \bold{a} \cdot \bold{b} = \| \bold a\|\|\bold b\| \cos \theta

    Here \bold a and \bold b are vectors and \theta is the angle between them. (Hint: one of your vectors should be lying in the plane )


    b
    ) Given the two planes
    E : . = 2.x + 3x2 + 7x3 + 2 = 0, E : x=(3,0,0)+ s(-2,0,0)+t(3,0,1)
    find the line of intersection and the angle between them.
    The normal vector for E' is given by \left< -2, 0, 0 \right> \times \left< 3, 0 , 1 \right>.

    The angle between the planes is the same as the angle between their normal vectors. Use the formula I gave you just above.

    As for the line of intersection, we can find it if we find a point on the line and a vector in the direction of the line. The latter can be found by taking the cross product of the two normal vectors (Why?). For the former, set x_3 = 0 in both planes. This will tell you on what lines the planes intersect with the xy-plane. Then just find the intersection point of those two lines (remember, the z-coordinate of those points is 0), that will give you a point on the line of intersection.

    Now see if you can get anywhere with those hints
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  3. #3
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    i understand how to do a) but im still not sure about b) im pretty sure im meant to convert the parametric equation (E') into a normal equation in the form of E but im not sure how i got about doing it.
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