Dot product Facts for Kids

In mathematics, the dot product is an operation that takes two vectors as input, and that returns a scalar number as output. The number returned is dependent on the length of both vectors, and on the angle between them. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar product emphasizes the scalar (rather than vector) nature of the result.

In three-dimensional space, the dot product contrasts with the cross product, which produces a vector as result.

Definition
Physics
Properties
Triple product expansion
Generalization
- Generalization to tensors
Related pages
Images for kids
See also

Definition

The dot product of two vectors a = [a₁, a₂, ..., a_n] and b = [b₁, b₂, ..., b_n] is defined as:

\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n

where Σ denotes summation notation (the sum of all the terms) and n is the dimension of the vector space.

In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd. The same way, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf. For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is

[1, 3, -5] \cdot [4, -2, -1] = (1 \times 4) + (3 \times (-2)) + ((-5) \times (-1)) = (4) - (6) + (5) = 3.

Physics

In physics, magnitude is a scalar in the physical sense, in that.it is a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. For example:

Mechanical work is the dot product of force and displacement vectors.
Magnetic flux is the dot product of the magnetic field and the area vectors.
Volumetric flow rate is the dot product of the fluid velocity and the area vectors.

Properties

The following properties hold if a, b, and c are real vectors and r is a scalar.

The dot product is commutative:

\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.

The dot product is distributive over vector addition:

\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.

The dot product is bilinear:

\mathbf{a} \cdot (r\mathbf{b} + \mathbf{c}) = r(\mathbf{a} \cdot \mathbf{b}) +(\mathbf{a} \cdot \mathbf{c}).

When multiplied by a scalar value, dot product satisfies:

(c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = (c_1c_2) (\mathbf{a} \cdot \mathbf{b})

(these last two properties follow from the first two).

Two non-zero vectors a and b are perpendicular if and only if a • b = 0.

Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law:

If a • b = a • c and a ≠ 0, then we can write: a • (b − c) = 0 by the distributive law; the result above says this just means that a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠ c.

Provided that the basis is orthonormal, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed. The above mentioned geometric interpretation relies on this property. In other words, for an orthonormal space with any number of dimensions, the dot product is invariant under a coordinate transformation based on an orthogonal matrix. This corresponds to the following two conditions:

The new basis is again orthonormal (that is, orthonormal expressed in the old one).
The new base vectors have the same length as the old ones (that is, unit length in terms of the old basis).

If a and b are functions, then the derivative of a • b is a' • b + a • b'.

Triple product expansion

This is a very useful identity (also known as Lagrange's formula) involving the dot- and cross-products. It is written as

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})

which is easier to remember as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula is commonly used to simplify vector calculations in physics.

Generalization

The inner product generalizes the dot product to abstract vector spaces and is usually denoted by $\langle\mathbf{a}\, , \mathbf{b}\rangle$ . Due to the geometric interpretation of the dot product, the norm ||a|| of a vector a in such an inner product space is defined as

\|\mathbf{a}\| = \sqrt{\langle\mathbf{a}\, , \mathbf{a}\rangle}

such that it generalizes length, and the angle θ between two vectors a and b by

\cos{\theta} = \frac{\langle\mathbf{a}\, , \mathbf{b}\rangle}{\|\mathbf{a}\| \, \|\mathbf{b}\|}.

In particular, two vectors are considered orthogonal if their inner product is zero

\langle\mathbf{a}\, , \mathbf{b}\rangle = 0.

For vectors with complex entries, using the given definition of the dot product would lead to quite different geometric properties. For instance, the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in turn would have severe consequences for notions like length and angle. Many geometric properties can be salvaged, at the cost of giving up the symmetric and bilinear properties of the scalar product, by alternatively defining

\mathbf{a}\cdot \mathbf{b} = \sum{a_i \overline{b_i}}

where b_i is the complex conjugate of b_i. Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, this scalar product is not linear in b (but rather conjugate linear), and the scalar product is not symmetric either, since

\mathbf{a} \cdot \mathbf{b} = \overline{\mathbf{b} \cdot \mathbf{a}}

This type of scalar product is nevertheless quite useful, and leads to the notions of Hermitian form and of general inner product spaces.

The Frobenius inner product generalizes the dot product to matrices. It is defined as the sum of the products of the corresponding components of two matrices having the same size.

Generalization to tensors

The dot product between a tensor of order n and a tensor of order m is a tensor of order n+m-2. The dot product is worked out by multiplying and summing across a single index in both tensors. If $\mathbf{A}$ and $\mathbf{B}$ are two tensors with element representation $A_{ij\dots}^{k\ell\dots}$ and $B_{mn\dots}^{p{\dots}i}$ the elements of the dot product $\mathbf{A} \cdot \mathbf{B}$ are given by

A_{ij\dots}^{k\ell\dots}B_{mn\dots}^{p{\dots}i} = \sum_{i=1}^n A_{ij\dots}^{k\ell\dots}B_{mn\dots}^{p{\dots}i}

This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices.

Occasionally, a double dot product is used to represent multiplying and summing across two indices. The double dot product between two 2nd order tensors is a scalar.

Cauchy–Schwarz inequality
Cross product
Matrix multiplication

Intuitive explanation video 1 and video 2 from online Interactive 3D graphics course

Images for kids

Vector components in an orthonormal basis

Dot product facts for kids

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