In mathematics and physics, a vector is (informally) a quantity with both magnitude and direction. A vector is a tensor of rank one. Vectors are commonly used (usually unknowingly) in everyday life; for instance, "five miles west" is a vector. Common vectors include position, velocity, and acceleration. A more formal definition of a vector is an element of a vector space .
Vectors are usuallu denoted in either rectangular or polar (or by extension, cylindrical or spherical) form. Rectangular form involves giving the magnitude of the vector in each dimension. For example, the vector with a magnitude of three in the x-direction, two in the y-direction and four in the z-direction can be denoted with ordered set notation, matrix notation, or unit vector notation:
For a vector in rectangular form, the magnitude is simply found with the Pythagorean theorem (the square root of the sum of the squares of the components). However, in some cases, vectors are given in terms of another basis rather than the unit vectors.
Vectors can also be denoted in polar form, where the magnitude of the vector is given along with the angle from a given axis.
- Vector addition: Vectors can be added by simply adding the components in each direction.
- Scalar multiplication: Vectors can be multiplied by scalars by multiplying each component of the vector by said scalar.
- Vector multiplication: This can be done in two ways, yielding either the dot or cross product, or, more generally, the inner or outer product.
- Triple product: Again, this can yield either the scalar triple product or the vector triple product.
- The zero vector, which has a magnitude of zero and an undefined direction
- A unit vector, which has a magnitude of 1 and is typically used to specify direction