Results 1 to 6 of 6

Math Help - Using vectors to find the volume of a pyramid

  1. #1
    Newbie
    Joined
    Mar 2011
    Posts
    8

    Using vectors to find the volume of a pyramid

    Hi,

    So here's my problem :

    The equation of a plane is 3x+2y-6z=12

    Which meets the x,y,and z-axis at points A, B, and C, which I calculated to be (4,0,0), (0,6,0), and (0,0,-2) respectively (I think I'm correct).

    Now I have to find the volume of the pyramid OABC. I'm guessing it's a triangle based pyramid with the peak at O (origin). I already found the area of the base to be 14 using the fact that A=1/2*magnitude(AC x AB).

    Now I'm stuck on how to find the height of the pyramid?

    EDIT: Or actually what just came to mind is the fact that the plane is 12 units from the origin can I use that fact as the height? (might have just solved it /facepalm/)

    Any help will be appreciated. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Does your syllabus cover the following formula?


    V=\dfrac{1}{6}\mod \begin{vmatrix} {x_1}&{y_1}&{z_1}&{1}\\{x_2}&{y_2}&{z_2}&{1}\\{x_3  }&{y_3}&{z_3}&{1}\\{x_4}&{y_4}&{z_4}&{1}\end{vmatr  ix}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2011
    Posts
    8
    I don't think so, I mean I've never seen it in our data booklet (IB) or my teachers notes.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Then, use that the height of the pyramid is 2 .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,792
    Thanks
    1532
    No, Fernando, not with FinnSkies's choice of base. If you take the base to be the triangle with vertices (4, 0, 0), 0,6,0), and (0,0,-2) then the height is the perpendicular distance from that plane to (0, 0, 0). That plane can be written as \frac{x}{4}+ \frac{y}{6}- \frac{z}{2}= 1 and so \frac{1}{4}\vec{i}+ \frac{1}{6}\vec{j}- \frac{1}{2}\vec{k} is normal to the plane. The line given by x= (1/4)t, y= (1/6)t, z= -(1/2)t is parallel to that vector and passes through (0, 0, 0). It intersects the plane when (1/16)t+ (1/36)t+ (1/4)t= \frac{49}{144}t= 1. t= 144/49 so that the point of intersection is (36/49, 4/49, -72/49). The height of the pyramid is the distance from (0, 0, 0) to that point, \frac{4}{7}\sqrt{406}.

    Of course, it is simpler to use one of the other planes as base. If we take the triangle with vertices at (0, 0, 0), (4, 0, 0), and (0, 6, 0) as base, then, because the third vertex, (0, 0, -2), is on the z-axis, perpendicular to that plane, the height is just 2.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by HallsofIvy View Post
    No, Fernando, not with FinnSkies's choice of base.

    Oops!, I misread the question.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. volume of a pyramid (not so easy)
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 6th 2010, 04:32 PM
  2. Volume of Pyramid..
    Posted in the Geometry Forum
    Replies: 4
    Last Post: March 23rd 2010, 02:59 AM
  3. Volume of a Pyramid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 27th 2010, 08:08 PM
  4. Volume of a pyramid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 12th 2009, 08:26 PM
  5. Volume of a pyramid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 1st 2009, 07:57 PM

Search Tags


/mathhelpforum @mathhelpforum