Formulas about vectors in three-dimensional Euclidean space
The following are important identities in vector algebra. Identities that involve the magnitude of a vector , or the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension. Identities that use the cross product (vector product) A×B are defined only in three dimensions.[1] (There is a seven-dimensional cross product, but the identities do not hold in seven dimensions.)
Magnitudes
The magnitude of a vector A can be expressed using the dot product:
In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem:
Inequalities
- The Cauchy–Schwarz inequality:
- The triangle inequality:
- The reverse triangle inequality:
Angles
The vector product and the scalar product of two vectors define the angle between them, say θ:[1][2]
To satisfy the right-hand rule, for positive θ, vector B is counter-clockwise from A, and for negative θ it is clockwise.
The Pythagorean trigonometric identity then provides:
If a vector A = (Ax, Ay, Az) makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then:
and analogously for angles β, γ. Consequently:
with unit vectors along the axis directions.
Areas and volumes
The area Σ of a parallelogram with sides A and B containing the angle θ is:
which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram. That is:
(If A, B are two-dimensional vectors, this is equal to the determinant of the 2 × 2 matrix with rows A, B.) The square of this expression is:[3]
where Γ(A, B) is the Gram determinant of A and B defined by:
In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A, B, C is given by the Gram determinant of the three vectors:[3]
Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product below.
This process can be extended to n-dimensions.
Addition and multiplication of vectors
- Commutativity of addition: .
- Commutativity of scalar product: .
- Anticommutativity of cross product: .
- Distributivity of multiplication by a scalar over addition: .
- Distributivity of scalar product over addition: .
- Distributivity of vector product over addition: .
- Scalar triple product: .
- Vector triple product: .
- Jacobi identity: .
- Binet-Cauchy identity: .
- Lagrange's identity: .
- Vector quadruple product:[4][5] .
- In 3 dimensions, a vector D can be expressed in terms of basis vectors {A,B,C} as:[6]
- .
See also
References