Results 1 to 7 of 7

Math Help - angle between planes

  1. #1
    tak
    tak is offline
    Junior Member
    Joined
    Apr 2008
    Posts
    29

    angle between planes

    x+2y-z=4
    2x+y+z=8

    calculate the angle between the planes
    i got 47 degree correct

    obtain the unit vector in the direction of the line of intersection of the two planes? I'm not too sure about this problem
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Apr 2008
    From
    United Kingdom
    Posts
    20
    Okay, so you have your two planes

    1: x+2y-z=4
    2: 2x+y+z=8

    I assume you've figured out by now that the normal to plane 1 is going to be in the direction (1, 2, -1) and the normal to plane 2 is going to be in the direction (2, 1, 1). I guess you then used the scalar (dot) product between them to work out what the angle was.

    The next bit: Unit vector in the line of intersection.

    Well, the unit bit just means it has a total length of 1 so let's worry about that later. First of all, work out what direction it's in.

    How do we do this? Think about your two planes. If you're having trouble visualising it, hold up some sheets of A4 to help you. Each plane has its own normal coming out of it at 90 degrees. The line of intersection is in BOTH planes, right? (fairly obviously, since it has to be in both for it to be the line of intersection)

    If a line is in a plane, then it's perpendicular to the normal of that plane. If it's in both planes, then it must be perpendicular to both normals. Again, that has to be true, by definition pretty much. Think about that for a minute and convince yourself that it's right.

    If you want to generate a vector perpendicular to two other vectors, how do you do it? That's right. The vector (or "cross") product.

    So it'll be in direction (1, 2, -1) CROSS (2, 1, 1).

    Once you know the direction, you just have to divide by the total magnitude to make sure it has length 1.

    Hope this helps.
    Last edited by Fedex; April 24th 2008 at 03:44 PM. Reason: clarifying a point
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    Quote Originally Posted by tak View Post
    x+2y-z=4
    2x+y+z=8

    calculate the angle between the planes
    i got 47 degree correct

    obtain the unit vector in the direction of the line of intersection of the two planes? I'm not too sure about this problem
    A vector in the direction of the line of intersection of two planes can be obtained by taking the cross-product of the normal vectors to the planes; so compute [1,2,-1]x [2,1,1] and then find the magnitude of this vector and divide by the magnitude to obtain the unit vector.

    Edit: Fedex beat me to it, and he has a nice explanation.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    tak
    tak is offline
    Junior Member
    Joined
    Apr 2008
    Posts
    29

    hmmm

    I go sought out my thoughts
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by tak View Post
    x+2y-z=4
    2x+y+z=8

    calculate the angle between the planes
    i got 47 degree correct

    obtain the unit vector in the direction of the line of intersection of the two planes? I'm not too sure about this problem
    isnt it arcos\bigg(\frac{n_1\cdot{n_2}}{|n_1||n_2|}\bigg)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    tak
    tak is offline
    Junior Member
    Joined
    Apr 2008
    Posts
    29

    i have a question

    why do we need the dot product to write the vector equation
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Apr 2008
    From
    United Kingdom
    Posts
    20
    I think Mathstud was reminding you how to calculate the angle between the two planes.

    If you call the normal to plane 1 n1 and the normal to plane 2 n2, then

    n1 DOT n2 = |n1||n2|cos(theta)

    Rearranging that will give you the angle between the normals as Mathstud showed. The angle between the normals must be the angle between the planes too (if you think about that by holding up pieces of paper and stuff, you'll see that that's true).

    I don't know if your answer is right, by the way. I can't work out inverse cosines in my head :-D I'll let you check it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Angle between planes and line of intersection of planes.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 6th 2011, 01:08 PM
  2. Bearing and angle between planes
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: March 10th 2010, 11:24 AM
  3. Replies: 0
    Last Post: May 22nd 2009, 08:11 PM
  4. Vector angle in multiple planes
    Posted in the Geometry Forum
    Replies: 2
    Last Post: January 23rd 2007, 08:59 PM
  5. angle between planes
    Posted in the Geometry Forum
    Replies: 1
    Last Post: October 31st 2006, 04:09 PM

Search Tags


/mathhelpforum @mathhelpforum