Define Dot Product: A.B = ||A|| ||B|| cos(theta)

[parshyaa]

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

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.

 

[blue_leaf77]

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 $(\mathbf v + \mathbf w,\mathbf z) = (\mathbf v,\mathbf z)+(\mathbf w,\mathbf z)$.

 

[Svein]

Here is a short explanation of the "dot product": https://en.wikipedia.org/wiki/Inner_product_space.

 

[robphy]

Possibly useful:
http://www.maa.org/sites/default/files/images/upload_library/4/vol6/Dray2/Dray.pdf "The Geometry of the Dot and Cross Products" by Tevian Dray & Corinne A. Manogue

 

[chiro]

Hey parshyaa.

The cosine rule is done for the general proof and one uses the results for length in arbitrary R^n.