Dot Product

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The dot product, or scalar product, is a type of vector multiplication in the Euclidean spaces, and is widely used in many areas of mathematics and physics. In ${\displaystyle {\mathbb{ℝ}}^3}$ there is another type of multiplication called the cross product ( or vector product), but it is only defined and makes sense in general for this particular vector space. Both the dot product and the cross product are widely used in in the study of optics, mechanics, electromagnetism, and gravitational fields, for example.

Definition[edit]

Given two vectors, A = (A₁, ... ,A_n) and B = (B₁, ... ,B_n) in ${\displaystyle {\mathbb{ℝ}}^n}$ with ${\displaystyle 1\le n\le 3}$, the dot product is defined as the product of the magnitude of A, the magnitude of B and the cosine of the smaller angle between them.

A • B = |A||B|cosθ_AB

In cartesian coordinates of dimension n>3 it has to be defined in a different way because it is no longer possible to visualize an "angle" between two vectors and there is no "natural" definition for it. In fact, in this case the contrary is true: it is the inner product which is defined directly while the notion of an angle is derived from this definition. For n>3, the dot product between A and B is defined as:

A • B = A₁B₁ + A₂B₂ + ... + A_nB_n

and the cosine of the angle θ_AB between A and B is then defined as

${\displaystyle \cos {\theta }_{AB}=\frac{\mathbf{𝐀}\cdot \mathbf{𝐁}}{|\mathbf{𝐀}||\mathbf{𝐁}|},}$

where ${\displaystyle |\mathbf{𝐀}|=(\mathbf{𝐀}\cdot \mathbf{𝐀}{)}^{1/2}}$ and ${\displaystyle |\mathbf{𝐁}|=(\mathbf{𝐁}\cdot \mathbf{𝐁}{)}^{1/2}}$ are, respectively, the magnitudes of A and B.

The dot product is a scalar, not another vector (unlike the cross product in ${\displaystyle {\mathbb{ℝ}}^3}$), and it obeys the following laws:

1. A • B = B • A (commutativity or symmetry)
2. (A₁ + A₂) • B = A₁ • B + A₂ • B (distributivity)
3. (cA) • B= A • (cB)=c(A • B) for any real number c

The first and second laws also imply that A • (B₁ + B₂) = A • B₁ + A • B₂

Two vectors with zero dot product are said to be orthogonal to each other.

Use in calculating work in physics[edit]

In mechanics, when a constant force F is applied over a straight displacement L, the work performed is FL cosθ_FL, more compactly written as the dot product below. Note in the special case the force and displacement are parallel, work = force ${\displaystyle \times }$ distance.

Work = F • L (linear motion)

Work = ${\displaystyle \int }$F • ${\displaystyle d}$L (non-linear motion)