Dot product

In mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the orthonormal Euclidean space.

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 and n is the dimension of the vectors.

For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is

For two complex column vectors the dot product is defined as

$mathbf\{a\}cdot\; mathbf\{b\}\; =\; sum\{overline\{b\_i\}\; a\_i\}$

where

$overline\{b\_i\}$

is the complex conjugate of $b\_i$; further note that in the complex case

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

The dot product is typically applied to vectors from orthonormal vector spaces. Its generalization to non-orthonormal vector spaces is described below.

Conversion to matrix multiplication

Using matrix multiplication and treating the vectors as n×1 matrices (i.e. "column matrices" or "column vectors"), the dot product can also be written as:

$mathbf\{a\}\; cdot\; mathbf\{b\}\; =\; mathbf\{a\}^T\; mathbf\{b\}\; ,$

where a^T denotes the transpose of the matrix a, and in this specific case, since a is a column matrix, the transpose of a is a "row matrix" or "row vector" (1×n matrix).

For instance, the dot product of the two above-mentioned three-dimensional vectors is equivalent to the product of a 1×3 matrix by a 3×1 matrix (which, by virtue of the matrix multiplication, results in a 1×1 matrix, i.e., a scalar):

$begin\{bmatrix\}$

   1&3&-5

end{bmatrix}begin{bmatrix}

   4-2-1

end{bmatrix} = begin{bmatrix}

   3

end{bmatrix}.

Geometric interpretation

In Euclidean geometry, the dot product, length, and angle are related. For a vector a, the dot product a · a is the square of the length of a, or

$|mathbf\{a\}|\; =\; sqrt\{mathbf\{a\}\; cdot\; mathbf\{a\}\}$

where |a| denotes the length (magnitude) of a. More generally, if b is another vector

$mathbf\{a\}\; cdot\; mathbf\{b\}\; =\; |mathbf\{a\}|\; ,\; |mathbf\{b\}|\; cos\; theta\; ,$

where

|a| and |b| denote the length of a and b

θ is the angle between them.

Thus, given two vectors, the angle between them can be found by rearranging the above formula:

$theta\; =\; arccos\; left(frac\; \{bold\{a\}cdotbold\{b\}\}$right).> As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero:

$text\{If\; \}\; mathbf\{a\}\; perp\; mathbf\{b\}\; text\{,\; then\; \}\; mathbf\{a\}\; cdot\; mathbf\{b\}\; =\; 0.$

Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length. This property provides a simple method to test the condition of orthogonality.

Sometimes these properties are also used for defining the dot product, especially in 2 and 3 dimensions; this definition is equivalent to the above one. For higher dimensions the formula can be used to define the concept of angle.

The geometric properties rely on the basis being orthonormal, i.e. composed of vectors perpendicular to each other and having unit length.

Scalar projection

If both a and b have length one (i.e. they are unit vectors), their dot product simply gives the cosine of the angle between them.

If only b is a unit vector, then the dot product a · b gives |a| cos(θ), i.e. the magnitude of the projection of a in the direction of b, with a minus sign if the direction is opposite. This is called the scalar projection of a onto b, or scalar component of a in the direction of b (see figure). This property of the dot product has several useful applications (for instance, see next section).

Rotation

A rotation of the orthonormal basis in terms of which vector a is represented is obtained with a multiplication of a by a rotation matrix R. This matrix multiplication is just a compact representation of a sequence of dot products.

For instance, let

• B₁ = {x, y, z} and B₂ = {u, v, w} be two different orthonormal bases of the same space R³, with B₂ obtained by just rotating B₁,
• a₁ = (a_x, a_y, a_z) represent vector a in terms of B₁,
• a₂ = (a_u, a_v, a_w) represent the same vector in terms of the rotated basis B₂,
• u₁, v₁, w₁ be the rotated basis vectors u, v, w represented in terms of B₁.

Then the rotation from B₁ to B₂ is performed as follows:

$bold\; a\_2\; =\; bold\{Ra\}\_1\; =$

begin{bmatrix} u_x & u_y & u_z v_x & v_y & v_z w_x & w_y & w_z end{bmatrix} begin{bmatrix} a_x a_y a_z end{bmatrix} = begin{bmatrix} bold u_1cdotbold a_1 bold v_1cdotbold a_1 bold w_1cdotbold a_1 end{bmatrix} = begin{bmatrix} a_u a_v a_w end{bmatrix} .

Notice that the rotation matrix R is assembled by using the rotated basis vectors u₁, v₁, w₁ as its rows, and these vectors are unit vectors. By definition, Ra₁ consists of a sequence of dot products between each of the three rows of R and vector a₁. Each of these dot products determines a scalar component of a in the direction of a rotated basis vector (see previous section). If a₁ is a row vector, rather than a column vector, then R must contain the rotated basis vectors in its columns, and must post-multiply a₁:

$bold\; a\_2\; =\; bold\; a\_1\; bold\; R\; =$

begin{bmatrix} a_x & a_y & a_z end{bmatrix} begin{bmatrix} u_x & v_x & w_x u_y & v_y & w_y u_z & v_z & w_z end{bmatrix} = begin{bmatrix} bold u_1cdotbold a_1 & bold v_1cdotbold a_1 & bold w_1cdotbold a_1 end{bmatrix} = begin{bmatrix} a_u & a_v & a_w end{bmatrix} .

The dot product in physics

In physics, magnitude is a scalar in the physical sense, i.e. 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. The formula in terms of coordinates is evaluated with not just numbers, but numbers times units. Therefore, although it relies on the basis being orthonormal, it does not depend on scaling.

Example:

• Mechanical work is the dot product of force and displacement

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 not associative, however with the help of the matrix-multiplication one can derive:

$(mathbf\{a\}\; cdot\; mathbf\{b\})\; cdot\; mathbf\{c\}\; =\; (mathbf\{c\}\; mathbf\{b\}^T)\; mathbf\{a\}.$

The dot product is bilinear:

$mathbf\{a\}\; cdot\; (rmathbf\{b\}\; +\; mathbf\{c\})$

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

When multiplied by a scalar value, dot product satisfies:

$(c\_1mathbf\{a\})\; cdot\; (c\_2mathbf\{b\})\; =\; (c\_1c\_2)\; (mathbf\{a\}\; cdot\; mathbf\{b\})$

(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; and from the previous result above:

If a is perpendicular to (b - c), we can have (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 (i.e., it is orthonormal expressed in the old one)
• the new base vectors have the same length as the old ones (i.e., unit length in terms of the old basis)

Derivative

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

$mathbf\{a\}\; times\; (mathbf\{b\}\; times\; mathbf\{c\})\; =\; mathbf\{b\}(mathbf\{a\}cdotmathbf\{c\})\; -\; mathbf\{c\}(mathbf\{a\}cdotmathbf\{b\})$

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.

Proof of the geometric interpretation

Note: This proof is shown for 3-dimensional vectors, but is readily extendable to n-dimensional vectors.

Consider a vector

$mathbf\{v\}\; =\; v\_1\; mathbf\{i\}\; +\; v\_2\; mathbf\{j\}\; +\; v\_3\; mathbf\{k\}.\; ,$

Repeated application of the Pythagorean theorem yields for its length v

$v^2\; =\; v\_1^2\; +\; v\_2^2\; +\; v\_3^2.\; ,$

But this is the same as

$mathbf\{v\}\; cdot\; mathbf\{v\}\; =\; v\_1^2\; +\; v\_2^2\; +\; v\_3^2,\; ,$

so we conclude that taking the dot product of a vector v with itself yields the squared length of the vector. Lemma 1:$mathbf\{v\}\; cdot\; mathbf\{v\}\; =\; v^2.\; ,$

Now consider two vectors a and b extending from the origin, separated by an angle θ. A third vector c may be defined as

$mathbf\{c\}\; stackrel\{mathrm\{def\}\}\{=\}\; mathbf\{a\}\; -\; mathbf\{b\}.\; ,$

creating a triangle with sides a, b, and c. According to the law of cosines, we have

$c^2\; =\; a^2\; +\; b^2\; -\; 2\; ab\; cos\; theta.\; ,$

Substituting dot products for the squared lengths according to Lemma 1, we get

$$

mathbf{c} cdot mathbf{c} = mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} - 2 ab costheta. ,                   (1) But as c ≡ a − b, we also have

$$

mathbf{c} cdot mathbf{c} = (mathbf{a} - mathbf{b}) cdot (mathbf{a} - mathbf{b}) ,, which, according to the distributive law, expands to

$$

mathbf{c} cdot mathbf{c} = mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} -2(mathbf{a} cdot mathbf{b}). ,                     (2) Merging the two c • c equations, (1) and (2), we obtain

$$

mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} -2(mathbf{a} cdot mathbf{b}) = mathbf{a} cdot mathbf{a} + mathbf{b} cdot mathbf{b} - 2 ab costheta. , Subtracting a • a + b • b from both sides and dividing by −2 leaves

$mathbf\{a\}\; cdot\; mathbf\{b\}\; =\; ab\; costheta.\; ,$

Q.E.D.

Generalization

The inner product generalizes the dot product to abstract vector spaces and is normally denoted by <a , b>. Due to the geometric interpretation of the dot product the norm

.

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

$langlemathbf\{a\},\; ,\; mathbf\{b\}rangle\; =\; 0.$

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.

Matrix representation

An inner product can be represented using a square matrix and matrix multiplication. For example, given two vectors

$$

mathbf{a} = begin{bmatrix} a_u a_v a_w end{bmatrix}, qquad mathbf{b} = begin{bmatrix} b_u b_v b_w end{bmatrix}

with respect to the basis set $mathrm\{S\}$

$$

mathrm{S} = { mathbf{u}, mathbf{v} ,mathbf{w} } = left{ begin{bmatrix} u_1 u_2 u_3 end{bmatrix}, begin{bmatrix} v_1 v_2 v_3 end{bmatrix}, begin{bmatrix} w_1 w_2 w_3 end{bmatrix} right}

any inner product can be represented as follows:

$$

langle mathbf{a}, , mathbf{b} rangle = mathbf{a}^T mathbf{M} mathbf{b}

where M is a 3x3 matrix. Given the matrix of the inner products through S called C_S, M can be calculated by solving the following system of equations.

If the basis set S is composed of orthogonal unit vectors (orthonormal basis), then both C_S and M reduce to the identity matrix 1, and the inner product can be represented as a simple product between a row matrix and a column matrix:

$$

langle mathbf{a}, , mathbf{b} rangle = mathbf{a}^T mathbf{1} mathbf{b} = mathbf{a}^T mathbf{b}.

Thus, if the basis is orthonormal, the square matrix M is not necessary and the inner product coincides with the dot product as defined above:

$$

langle mathbf{a}, , mathbf{b} rangle = mathbf{a}^T mathbf{b} = mathbf{a} cdot mathbf{b}.

Example

Given a basis set

$$

mathrm{S} = { mathbf{u}, mathbf{v} ,mathbf{w} } = left{ begin{bmatrix} 1 0 0 end{bmatrix}, begin{bmatrix} 1 1 0 end{bmatrix}, begin{bmatrix} 1 1 1 end{bmatrix} right}

and a matrix of the inner product through $mathrm\{S\}$

$$

mathbf{C}_S = begin{bmatrix}

       5 & 2 & 0 

       2 & 6 & 2 

       0 & 2 & 7

end{bmatrix}

we can set each element of C_S equal to the inner product of two of the basis vectors as follows

$$

mathbf{C}_S[i,j] = langle mathrm{S}[i],mathrm{S}[j] rangle

$$

mathbf{C}_S[0,0] = 5 = langle mathbf{u,u} rangle = begin{bmatrix} 1 & 0 & 0 end{bmatrix} mathrm{M} begin{bmatrix} 1 0 0 end{bmatrix}

$$

mathbf{C}_S[0,1] = 2 = langle mathbf{u,v} rangle = begin{bmatrix} 1 & 0 & 0 end{bmatrix} mathrm{M} begin{bmatrix} 1 1 0 end{bmatrix}

$$

   cdots

which gives nine equations and nine unknowns. Solving these equations yields

$$

mathbf{M} = begin{bmatrix}

       5 & -3 & -2 

       -3 & 7 & -2 

       -2 & -2 & 9

end{bmatrix}.

See also

• Cross product
• Matrix multiplication
• Cauchy–Schwarz inequality

External links

• Interactive GeoGebra Applet
• Java demonstration of dot product
• Another Java demonstration of dot product
• Explanation of dot product including with complex vectors
• "Dot Product" by Bruce Torrence, The Wolfram Demonstrations Project, 2007.

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