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Dot Product


DotProduct

The dot product can be defined for two vectors X and Y by

 X·Y=|X||Y|costheta,
(1)

where theta is the angle between the vectors and |X| is the norm. It follows immediately that X·Y=0 if X is perpendicular to Y. The dot product therefore has the geometric interpretation as the length of the projection of X onto the unit vector Y^^ when the two vectors are placed so that their tails coincide.

By writing

A_x=Acostheta_A    B_x=Bcostheta_B
(2)
A_y=Asintheta_A    B_y=Bsintheta_B,
(3)

it follows that (1) yields

A·B=ABcos(theta_A-theta_B)
(4)
=AB(costheta_Acostheta_B+sintheta_Asintheta_B)
(5)
=Acostheta_ABcostheta_B+Asintheta_ABsintheta_B
(6)
=A_xB_x+A_yB_y.
(7)

So, in general,

X·Y=sum_(i=1)^(n)x_iy_i
(8)
=x_1y_1+...+x_ny_n.
(9)

This can be written very succinctly using Einstein summation notation as

 X·Y=x_iy_i.
(10)

The dot product is implemented in the Wolfram Language as Dot[a, b], or simply by using a period, a . b.

The dot product is commutative

 X·Y=Y·X,
(11)

and distributive

 X·(Y+Z)=X·Y+X·Z.
(12)

The associative property is meaningless for the dot product because (a·b)·c is not defined since a·b is a scalar and therefore cannot itself be dotted. However, it does satisfy the property

 (rX)·Y=r(X·Y)
(13)

for r a scalar.

The derivative of a dot product of vectors is

 d/(dt)[r_1(t)·r_2(t)]=r_1(t)·(dr_2)/(dt)+(dr_1)/(dt)·r_2(t).
(14)

The dot product is invariant under rotations

A^'·B^'=A_i^'B_i^'
(15)
=a_(ij)A_ja_(ik)B_k
(16)
=(a_(ij)a_(ik))A_jB_k
(17)
=delta_(jk)A_jB_k
(18)
=A_jB_j
(19)
=A·B,
(20)

where Einstein summation has been used.

The dot product is also called the scalar product and inner product. In the latter context, it is usually written <a,b>. The dot product is also defined for tensors A and B by

 A·B=A^alphaB_alpha.
(21)

So for four-vectors a_mu and b_mu, it is defined by

a_mu·b_mu=a_mub^mu
(22)
=a^0b^0-a^1b^1-a^2b^2-a^3b^3
(23)
=a^0b^0-a·b,
(24)

where a·b is the usual three-dimensional dot product.


See also

Cross Product, Einstein Summation, Four-Vector Norm, Inner Product, Outer Product, Perp Dot Product, Vector, Vector Multiplication, Wedge Product Explore this topic in the MathWorld classroom

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References

Arfken, G. "Scalar or Dot Product." §1.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 13-18, 1985.Jeffreys, H. and Jeffreys, B. S. "Scalar Product." §2.06 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 65-67, 1988.

Referenced on Wolfram|Alpha

Dot Product

Cite this as:

Weisstein, Eric W. "Dot Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DotProduct.html

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