Difference between revisions of "Dot product"
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| − | + | In the <math>n</math>-dimensional Euclidean vector space <math>\mathbb{R}^n</math>, the '''dot product''' is defined for two [[vector]]s <math>\vec{x} = | |
| + | \langle x_1, \ldots, x_n \rangle</math> and <math>\vec{y} = \langle y_1, \ldots, y_n \rangle</math> as follows: | ||
| − | + | : <math>\vec{x} \cdot \vec{y} = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots +x_n y_n </math> | |
| − | Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction. | + | Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction. Also, unlike the cross product, the dot product is [[commutative]]. |
| − | + | For example, in 3-dimensional Euclidean space, let <math>\vec{x} = \langle 1, 2, 3 \rangle</math> and <math>\vec{y} = \langle 4, 5, 6\rangle</math>. | |
| + | Then we can calculate <math>\vec{x} \cdot \vec{y} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32</math>. | ||
| − | + | The dot product of any vector with itself is the square of its [[norm]]. | |
| − | === | + | In 2- and 3- dimensional Euclidean space, the relation between the dot product of two vectors and the angle between them is given by |
| + | |||
| + | : <math>\vec{x} \cdot \vec{y} = |\vec{x}||\vec{y}|\cos\theta</math> | ||
| + | where <math>|\cdot|</math> is the vector norm and <math>\theta</math> is the angle between the vectors. Note that the vectors are perpendicular | ||
| + | (or [[orthogonal]]) if and only if their dot product is zero. | ||
| + | |||
| + | This relationship can be extended to define the concept of the angle between vectors in higher-dimensional Euclidean vector spaces. Two vectors | ||
| + | in <math>\mathbb{R}^n</math> are defined to be [[orthogonal]] if their dot product is zero. | ||
| + | |||
| + | The dot product is the best-known example of an [[inner product]]. | ||
| + | |||
| + | == Application == | ||
| + | |||
| + | The dot product is useful in projecting one vector onto another, as in calculating the work done by applying a force to a particle. If you know the dot product of two vectors, then you can easily calculate the angle between the vectors. | ||
| + | |||
| + | The dot product can also be used to find other values through application of the [[Stokes' Theorem]] and other theorems. | ||
| + | |||
| + | ==See also== | ||
[[Cross product]] | [[Cross product]] | ||
| − | [[ | + | [[Category:Linear Algebra]] |
| − | [[ | + | [[Category:Vector Analysis]] |
| + | [[Category:Calculus]] | ||
| + | [[Category:Mathematics]] | ||
| + | [[Category:Physics]] | ||
Latest revision as of 03:41, July 5, 2018
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This article/section deals with mathematical concepts appropriate for late high school or early college. |
In the 
-dimensional Euclidean vector space 
, the dot product is defined for two vectors 
and 
as follows:
Unlike the cross product, the dot product is a scalar, not a vector, and has no direction. Also, unlike the cross product, the dot product is commutative.
For example, in 3-dimensional Euclidean space, let 
and 
.
Then we can calculate 
.
The dot product of any vector with itself is the square of its norm.
In 2- and 3- dimensional Euclidean space, the relation between the dot product of two vectors and the angle between them is given by
where 
is the vector norm and 
is the angle between the vectors. Note that the vectors are perpendicular
(or orthogonal) if and only if their dot product is zero.
This relationship can be extended to define the concept of the angle between vectors in higher-dimensional Euclidean vector spaces. Two vectors
in are defined to be orthogonal if their dot product is zero.
The dot product is the best-known example of an inner product.
Application
The dot product is useful in projecting one vector onto another, as in calculating the work done by applying a force to a particle. If you know the dot product of two vectors, then you can easily calculate the angle between the vectors.
The dot product can also be used to find other values through application of the Stokes' Theorem and other theorems.
