Dot product

${\frac {d}{dx}}\sin x=?\,$

This article/section deals with mathematical concepts appropriate for late high school or early college.

In the $n$-dimensional Euclidean vector space $\mathbb {R} ^{n}$, the dot product is defined for two vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x} = \langle x_1, \ldots, x_n \rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y} = \langle y_1, \ldots, y_n \rangle} as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x} \cdot \vec{y} = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots +x_n y_n }

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x} = \langle 1, 2, 3 \rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y} = \langle 4, 5, 6\rangle} . Then we can calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x} \cdot \vec{y} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32} .

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x} \cdot \vec{y} = |\vec{x}||\vec{y}|\cos\theta}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\cdot|} is the vector norm and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} 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