Dot Product (original) (raw)

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

Alphabetical Index New in MathWorld

The dot product can be defined for two vectors and by

| | (1) | | --------------------------------------------------------------------------------------------------------- | --- |

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

By writing

it follows that (1) yields

So, in general,

This can be written very succinctly using Einstein summation notation as

	(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

	(11)

and distributive

	(12)

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

	(13)

for a scalar.

The derivative of a dot product of vectorsis

	(14)

The dot product is invariant under rotations

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 . The dot product is also defined for tensors and by

	(21)

So for four-vectors and , it is defined by

where 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

Explore with Wolfram|Alpha

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." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DotProduct.html

Subject classifications