Results 1 to 5 of 5

Math Help - Angle between two 3D vectors with proper sign

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    21

    Angle between two 3D vectors with proper sign

    I have two vectors

    A = 2i + 3j + 4k
    B = 5i + 6j + 7k

    I wish to find out the angle between them with the proper sign.
    I read (Maths - Inverse Trigonometric Functions - Martin Baker) that the function atan2 is the right way to do it. But I don't find the method to go about it. I want to do it by hand not using some software or excel or something.
    Anyone throw some light on it or any other proper method.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,802
    Thanks
    1691
    Awards
    1
    Quote Originally Posted by ontherocks View Post
    I have two vectors
    A = 2i + 3j + 4k
    B = 5i + 6j + 7k
    I wish to find out the angle between them with the proper sign.
    The angle between is: \arccos \left( {\frac{{A \cdot B}}<br />
{{\left\| A \right\|\left\| B \right\|}}} \right).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    21
    Unfortunately, arccos doesn't give the angle "with proper sign".

    Some discussions on similar lines (http://www.gamedev.net/community/for...opic_id=503639)
    Last edited by ontherocks; December 17th 2009 at 10:10 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,802
    Thanks
    1691
    Awards
    1
    Quote Originally Posted by ontherocks View Post
    Unfortunately, arccos doesn't give the angle "with proper sign".
    You are completely mistaken about that.
    The angle between any two vectors is such 0\le\theta\le \pi .
    The angle between two vectors does not have a sign.
    There is no such a thing mathematically speaking.
    I read the discussion, someone there said the same thing.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,811
    Thanks
    701
    Hello, ontherocks!

    I have two vectors: . \begin{array}{ccc}\vec a &=& 2i + 3j + 4k \\ \vec b &=& 5i + 6j + 7k\end{array}

    I wish to find out the angle between them.
    Plato's formula is correct.

    . . \cos\theta \;=\;\frac{\vec a \cdot \vec b}{|\vec a||\vec b|}

    and it does give the proper sign!



    We have: . \begin{array}{ccc} \vec a &=& \langle 2,3,4\rangle \\ \vec b &=& \langle 5,6,7\rangle \end{array}

    Then: . \cos\theta \;=\;\frac{\langle2,3,4\rangle\cdot\langle5,6,7\ra  ngle}{\sqrt{2^2+3^2+4^2}\,\sqrt{5^2+6^2+7^2}} \;=\;\frac{10+18+28}{\sqrt{4+9+16}\,\sqrt{25+36+49  }} \;=\;\frac{56}{\sqrt{29}\,\sqrt{110}}

    Therefore: . \theta \;=\;\cos^{-1}\left(\frac{56}{\sqrt{29}\,\sqrt{110}}\right) \;\approx\;7.5^o


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    Suppose we have: . \vec v \:=\:\langle \text{-}5, \text{-}6,\text{-}7\rangle . . . the negative of \vec b.


    The angle between \vec a and \vec v is given by:

    . . \cos\alpha \;=\;\frac{\vec a\cdot \vec v}{|\vec a||\vec v|} \;=\;\frac{-10 - 18 - 28}{\sqrt{4+9+16}\,\sqrt{25+36+49}} \;=\;\frac{-56}{\sqrt{29}\,\sqrt{110}}


    Therefore: . \alpha \;=\;\cos^{-1}\left(\frac{-56}{\sqrt{29}\,\sqrt{110}}\right) \quad\Rightarrow\quad \alpha \:\approx\:172.5^o

    See?

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vectors and an angle
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 6th 2011, 09:24 AM
  2. Angle between two vectors
    Posted in the Math Software Forum
    Replies: 6
    Last Post: November 22nd 2010, 07:40 AM
  3. Angle between Vectors
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 26th 2009, 07:44 AM
  4. Angle from 3d Vectors
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: May 1st 2008, 06:49 AM
  5. angle between the vectors ..........
    Posted in the Calculus Forum
    Replies: 7
    Last Post: April 6th 2007, 09:04 AM

Search Tags


/mathhelpforum @mathhelpforum