1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cross and dot products?

  1. Jan 21, 2010 #1
    What is significance of the trig functions in both the cross and the dot product? I understand what the dot and cross products are, how they work, and what they give...but I don't understand why the dot product uses cosine and the cross product uses sine?
     
  2. jcsd
  3. Jan 21, 2010 #2
    Actually never mind. I'm reading some proofs and my trig skills are rusty, so it doesn't make any sense. I know how to use cross/dot products and what they do, so that's close enough for me.
     
  4. Jan 21, 2010 #3

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Let's assume you define the dot product as [itex]A\cdot B = A_xB_x+A_yB_y+A_zB_z[/itex]. You can easily show that rotations don't affect the dot product. If you have two vectors A and B and rotate the system so they are now A' and B', you'll get the same result for the dot product using either pair, i.e. [itex]A\cdot B=A'\cdot B'[/itex]. So you can always perform a rotation so that A points along the x-axis, so that A = |A|(1,0,0). The dot product will therefore equal [itex]A\cdot B = |A|B_{x}[/itex]. Now [itex]B_{x}[/itex] is just the projection of B onto the x-axis, which, using basic trig, is [itex]B_{x} = |B|\cos \theta[/itex], where [itex]\theta[/itex] is the angle B makes with the x-axis, which is also the angle between A and B. So you get [itex]A\cdot B = |A||B|\cos\theta[/itex].

    The dot product is a special case of what's called an inner product. The definition above is the inner product for plain old three-dimensional Euclidean space, but other spaces are characterized by having a different inner product. If you have two vectors A and B in such a space, you can use

    [tex]\cos\theta=\frac{\langle A,B\rangle}{\sqrt{\langle A,A\rangle}\sqrt{\langle B,B\rangle}}[/tex]

    where [itex]\langle A,B\rangle[/itex] is the inner product of A and B, to define angles in this space. In this case, the cosine is there by definition.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cross and dot products?
  1. Cross and dot product ! (Replies: 11)

  2. Dot and cross product (Replies: 2)

Loading...