Vector algebra relations

The following are important identities in vector algebra. Identities that involve the magnitude of a vector $\, \mathbf A\,$, 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.

Magnitudes

The magnitude of a vector A can be expressed using the dot product:

:$\, \mathbf A \, ^2 = \mathbf$

In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem:

:$\, \mathbf A \, ^2 = A_1^2 + A_2^2 +A_3^2$

Inequalities

*The Cauchy–Schwarz inequality: $\mathbf \cdot \mathbf \le \left\, \mathbf A \right\, \left\, \mathbf B \right\,$
*The triangle inequality: $\, \mathbf\, \le \, \mathbf\, + \, \mathbf\,$
*The reverse triangle inequality: $\, \mathbf\, \ge \Bigl, \, \mathbf\, - \, \mathbf\, \Bigr,$

Angles

The vector product and the scalar product of two vectors define the angle between them, say ''θ'':

:$\sin \theta =\frac \quad ( -\pi < \theta \le \pi )$
To satisfy the right-hand rule, for positive ''θ'', vector B is counter-clockwise from A, and for negative ''θ'' it is clockwise.
:$\cos \theta = \frac \quad ( -\pi < \theta \le \pi )$

The Pythagorean trigonometric identity then provides:

:$\left\, \mathbf\right\, ^2 +(\mathbf \cdot \mathbf)^2 = \left\, \mathbf A \right\, ^2 \left\, \mathbf B \right\, ^2$

If a vector A = (''A_x, A_y, A_z'') makes angles ''α'', ''β'', ''γ'' with an orthogonal set of ''x-'', ''y-'' and ''z-''axes, then:

:$\cos \alpha = \frac = \frac \ ,$
and analogously for angles β, γ. Consequently:
:$\mathbf A = \left\, \mathbf A \right\, \left( \cos \alpha \ \hat + \cos \beta\ \hat + \cos \gamma \ \hat \right) ,$
with $\hat, \ \hat, \ \hat$ 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:
:$\Sigma = AB \sin \theta ,$
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:
:$\Sigma = \left\, \mathbf \times \mathbf \right\, = \sqrt \ .$

(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:
:$\Sigma^2 = (\mathbf)(\mathbf)-(\mathbf)(\mathbf)=\Gamma(\mathbf A,\ \mathbf B ) \ ,$
where Γ(A, B) is the Gram determinant of A and B defined by:

:$\Gamma(\mathbf A,\ \mathbf B )=\begin \mathbf & \mathbf \\ \mathbf & \mathbf \end \ .$

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:
:$V^2 =\Gamma ( \mathbf A ,\ \mathbf B ,\ \mathbf C ) = \begin \mathbf & \mathbf & \mathbf \\\mathbf & \mathbf & \mathbf\\ \mathbf & \mathbf & \mathbf \end \ ,$

Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product \det mathbf,\mathbf,\mathbf= , \mathbf,\mathbf,\mathbf, below.

This process can be extended to ''n''-dimensions.

Addition and multiplication of vectors

* Commutativity of addition: $\mathbf+\mathbf=\mathbf+\mathbf$.
* Commutativity of scalar product: $\mathbf\cdot\mathbf=\mathbf\cdot\mathbf$.
* Anticommutativity of cross product: $\mathbf\times\mathbf=\mathbf\times\mathbf$.
* Distributivity of multiplication by a scalar over addition: $c (\mathbf+\mathbf) = c\mathbf+c\mathbf$.
* Distributivity of scalar product over addition: $\left(\mathbf+\mathbf\right)\cdot\mathbf=\mathbf\cdot\mathbf+\mathbf\cdot\mathbf$.
* Distributivity of vector product over addition: $(\mathbf+\mathbf)\times\mathbf = \mathbf\times\mathbf+\mathbf\times\mathbf$.
* Scalar triple product: $\mathbf\cdot (\mathbf\times\mathbf)=\mathbf\cdot (\mathbf\times\mathbf)=\mathbf\cdot (\mathbf\times\mathbf) = , \mathbf\, \mathbf\,\mathbf, = \begin A_ & B_ & C_\\ A_ & B_ & C_\\ A_ & B_ & C_\end.$
* Vector triple product: $\mathbf\times (\mathbf\times\mathbf) = (\mathbf\cdot\mathbf )\mathbf- (\mathbf\cdot\mathbf)\mathbf$.
* Jacobi identity: $\mathbf\times (\mathbf\times\mathbf )+\mathbf\times (\mathbf\times\mathbf )+ \mathbf\times (\mathbf\times\mathbf )= \mathbf 0 .$
* Binet-Cauchy identity: $\mathbf\left(\mathbf\times\mathbf\right)=\left(\mathbf\cdot\mathbf\right) \left(\mathbf\cdot\mathbf\right) - \left(\mathbf\cdot\mathbf\right) \left(\mathbf\cdot\mathbf\right) .$
* Lagrange's identity: $, \mathbf \times \mathbf, ^2 = (\mathbf \cdot \mathbf) (\mathbf \cdot \mathbf)-(\mathbf \cdot \mathbf)^2$.
* Vector quadruple product:This formula is applied to spherical trigonometry by $(\mathbf \times \mathbf) \times (\mathbf \times \mathbf) \ =\ , \mathbf\,\mathbf\, \mathbf, \,\mathbf\,-\,, \mathbf\,\mathbf\, \mathbf, \,\mathbf\ =\ , \mathbf\,\mathbf\, \mathbf, \,\mathbf\,-\,, \mathbf\, \mathbf\,\mathbf, \,\mathbf.$
* A consequence of the previous equation: $, \mathbf\, \mathbf\,\mathbf, \,\mathbf= (\mathbf\cdot\mathbf )\left(\mathbf\times\mathbf\right)+\left(\mathbf\cdot\mathbf\right)\left(\mathbf\times\mathbf\right)+\left(\mathbf\cdot\mathbf\right)\left(\mathbf\times\mathbf\right).$
*In 3 dimensions, a vector D can be expressed in terms of basis vectors as:$\mathbf D \ =\ \frac\ \mathbf A +\frac\ \mathbf B + \frac\ \mathbf C.$

See also

*Vector space
*Geometric algebra

Notes

References

{{Reflist

Mathematical identities
Mathematics-related lists