1. Limited time only! Sign up for a free 30min personal 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!

I Define dot product

  1. Jul 31, 2016 #1
    I have seen a proof for the formula of A.B =
    ||A|| ||B|| cos(theta)[ proof using the diagram and cosine rule]. In the proof they have assumed that distributive property of dot product is right. diagram is given below
    100px-Dot_product_cosine_rule.svg.png c.c =(a-b).(a-b) = a^2 +b^2 -2(a.b) [ here they used distributive law]
    • I have seen another proof for the distributive property of dot product. There they have assumed that A.B = ||A|| ||B|| cos(theta),And used projections. They have used the diagram as given below.
    And projection of vector B on A is ||B||cos(theta) = B.a^ ( a^ is a unit vector in the direction of a vector)[ this is possible if the formula of dot product is assumed to be right.
    • How they can do this , for proving dot product A.B = ||A|| ||B|| cos(theta) they have assumed distributive property to be right and for prooving distributive property they have assumed dot product to be right.
    Therefore I think that there will be a definition for dot product wether it is A.B = ||A|| ||B|| cos(theta) or A.B = a1a2 +b1b2 +c1c2 (component form). If its a definition then how they have defined it like this.
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Jul 31, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper

    The distributive property of inner product follows immediately from the basic definition of the standard inner product. Given two vectors ##\mathbf v## and ##\mathbf w##, their inner product is ##(\mathbf v, \mathbf w) = \overline v_1 w_1 + \overline v_2 w_2 + \ldots + \overline v_N w_N##. From this, it should be straight forward to see that ##(\mathbf v + \mathbf w,\mathbf z) = (\mathbf v,\mathbf z)+(\mathbf w,\mathbf z)##.
  4. Jul 31, 2016 #3


    User Avatar
    Science Advisor

  5. Jul 31, 2016 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

  6. Aug 1, 2016 #5


    User Avatar
    Science Advisor

    Hey parshyaa.

    The cosine rule is done for the general proof and one uses the results for length in arbitrary R^n.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted