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In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken ...
A Euclidean vector may possess a definite initial point and terminal point; such a condition may be emphasized calling the result a bound vector. [12] When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a free vector.
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds: The following relationship holds: a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b ...
Since the vector term of the vector bivector product the name dot product is zero when the vector is perpendicular to the plane (bivector), and this vector, bivector "dot product" selects only the components that are in the plane, so in analogy to the vector-vector dot product this name itself is justified by more than the fact this is the non ...
A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ^ (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space. [1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane.