Dot product

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For the abstract scalar product or inner product see inner product space

In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. It is the standard inner product of the Euclidean space.

The dot product of two vectors a = [a1, a2, … , an] and b = [b1, b2, … , bn] is by definition

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

where Σ denotes summation notation. For example, the dot product of two three-dimensional vectors [1, 3, −2] and [4, −2, −1] is

[1, 3, −2]·[4, −2, −1] = 1×4 + 3×(−2) + (−2)×(−1) = 0.

Using matrix multiplication and treating the row vectors as 1×n matrices, the dot product can also be written as

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

where bT denotes the transpose of the matrix b. Using the example from above, this would result in a 1×3 matrix (i.e. vector) multiplied by a 3×1 vector (which, by virtue of the matrix multiplication, results in a 1×1 matrix):

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 \begin{bmatrix}1&3&-2\end{bmatrix}\begin{bmatrix}4\\-2\\-1\end{bmatrix} = \begin{bmatrix}0\end{bmatrix}}


Geometric interpretation

In the Euclidean space there is a strong relationship between the dot product and lengths and angles. For a vector a, a·a is the square of its length, and if b is another vector

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

where a and b denote the length of a and b, and θ is the angle between them.

Since a·cos(θ) is the projection of a onto b, the dot product can be understood geometrically as the product of this projection with the length of b.

As the cosine of 90° is zero, the dot product of two perpendicular vectors is always zero. If a and b have length one (they are unit vectors), the dot product simply gives the cosine of the angle between them. Thus, given two vectors, the angle between them can be found by rearranging the above formula:

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 \cos{\theta} = \frac{\mathbf{a} \cdot \mathbf{b}}{a b}. }

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 vectors being perpendicular and having unit length: either we start with such a basis, or we use an arbitrary basis and define length and angle (including perpendicularity) with the above.

As the geometric interpretation shows, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed.

In other words, and more generally for any n, 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)

The dot product in physics

In physics, for a vector a, a·a is the square of its magnitude, and if b is another vector

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

where a and b denote the magnitude of a and b, and θ is the angle between them.

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:

Properties

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

The dot product is commutative:

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

The dot product is bilinear:

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

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

If b is a unit vector, then the dot product gives the magnitude of the projection of a in the direction b, with a minus sign if the direction is opposite. Decomposing vectors is often useful for conveniently adding them, e.g. in the calculation of net force in mechanics.

Generalization

The inner product generalizes the dot product to abstract vector spaces, it is normally denoted by <a, b>. 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

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 \|\mathbf{a}\| = \sqrt{<\mathbf{a}, \mathbf{a}>}} ,

such that it generalizes length, and the angle θ between two vectors a and b 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 \cos{\theta} = \frac{<\mathbf{a}, \mathbf{b}>}{\|\mathbf{a}\| \, \|\mathbf{b}\|}. }

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

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

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 \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

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 v^2 = v_1^2 + v_2^2 + v_3^2. \;}

But this is the same as

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 \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
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 \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

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 \mathbf{c} \equiv \mathbf{a} - \mathbf{b} \;} ,

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

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

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

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 \mathbf{c} \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} - 2 ab \cos\theta \; } .                   (1)

But as cab, we also have

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

which, according to the distributive law, expands to

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 \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

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 \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 \cos\theta \; } .

Subtracting a · a + b · b from both sides and dividing by −2 leaves

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

Q.E.D.

See also

External links

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