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 crucial in physics, as well as some mathematical fields. Vector calculus is the application of calculus to vector fields, and linear algebra is the study of vector spaces.

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 \mathbf v 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:

\mathbf{v}=(3,2,4)=\begin{bmatrix}3\\2\\4\\ \end{bmatrix}=3\mathbf{\hat{i}}+2\mathbf{\hat{j}}+4\mathbf{\hat{k}}

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 x,y,z 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 operations

Special vectors

  • 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

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