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Math Help - vector question

  1. #1
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    vector question

    Hi,
    How do you find the volume of a cone on a plane?I'm given the equation of the plane and two points- one on the circular area of the cone and the other on the top (at the pointy part of the cone). I have an idea how to solve it but i'm not 100%sure. I thought since we know the normal of the plane, we know the direction vector of the line that goes through the point on top of the cone,so we know the vector equation for it. Then we find the normal to that line that goes through the second point on the cone and find the intersection point between those 2 lines which will be right in the center of the circle.Khowing that point helps us to find the radius and the distance from that point to the point on top of the cone.Then I just use the formula to find the volume but I'm not sure if I'm right or not.Please help me )
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  2. #2
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    A picture is worth 1000 words, so here's 1000 words for you...

    The text of the original question would be helpful here.

    If I am visualizing correctly, you have the equation of a plane with normal vector N, the location of the apex of the cone, A, and a point on the circular edge of the cone, B, its circular area contained in the plane.

    If this is correct, you can find the equation of the line in the direction of N passing through A, and the point at which this line passes through the plane will be the center of its circular area, C. Therefore, the radius of the cone's circular area will be the distance BC, and the height of the cone will be AC.
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  3. #3
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    lol Anna prob goes to my school. BHS BRUINS ALL DAY.
    lol so I need the exact same question i guess i can give some detail.
    they never gave us the EXACT question. we were just told the details.
    1. Cone on a Plane
    2. Equation of the plane is given
    3. The top (point) of the cone is perpendicular to the plane
    4. Another point of the cone on the plane is given
    -I think it is on like the bottom left of the cone

    yea i think that is all
    then they just told us find the volume of the cone?
    anyone knows how to? would be veryyyyy helpful.
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  4. #4
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    Quote Originally Posted by Media_Man View Post
    The text of the original question would be helpful here.

    If I am visualizing correctly, you have the equation of a plane with normal vector N, the location of the apex of the cone, A, and a point on the circular edge of the cone, B, its circular area contained in the plane.

    If this is correct, you can find the equation of the line in the direction of N passing through A, and the point at which this line passes through the plane will be the center of its circular area, C. Therefore, the radius of the cone's circular area will be the distance BC, and the height of the cone will be AC.
    yeah you're right but how do I find the center of the circular area. How do you find the vector equation of the line that passes through B and intersects with the line that passes through A?
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  5. #5
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    How about a dry run?

    Plane: 5x+3y+2z=7 , normal N=(5,3,2)
    Apex of cone: (10,10,17)
    Point in plane on circular edge of cone: (1,0,1)

    The line perpendicular to the plane passing through A is of the form L(t)=(a,b,c)+t(5,3,2) for some starting point L(0)=(a,b,c). Let (a,b,c) be the point of intersection of the line and plane, so 5a+3b+2c=7. We then have: (a,b,\frac{7-5a-3b}2)+t(5,3,2)=(10,10,17). This is a 3x4 matrix with solution: a=-4.07,b=1.55,t=2.82 (sorry the numbers didn't come out pretty). This makes c=11.35. The point (a,b,c) is the center of the circular area of the cone by construction. Its distance from the apex (the cone's height) is 17.35 and the radius of the cone is the distance from this center to the point on the edge, 11.62. The volume is easy to compute from here.

    Like I said, the original text of the problem will be helpful.
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