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In Physics and in Vector Calculus , a spatial vector, or simply '''vector''', is a concept characterized by a Magnitude and a direction. A vector has properties that do not depend on the coordinate system used to describe it. However, a vector is often described by a fixed number of '''components''', each of which is dependent upon the particular coordinate system being used, such as Cartesian Coordinates , Spherical Coordinates or Polar Coordinates . A common example of a vector is Force — it has a magnitude and an orientation and multiple forces sum according to the Parallelogram Law . A spatial vector can be formally defined by its relationship to the spatial Coordinate System under rotations. Alternatively, it can be defined in a Coordinate -free fashion via a Tangent Space of a three-dimensional Manifold in the language of Differential Geometry . These definitions are discussed in more detail below. A spatial vector is a special case of a Tensor and is also analogous to a Four-vector in Relativity (and is sometimes therefore called a ''three-vector'' in reference to the three spatial dimensions, although this term also has another meaning for P-vectors of differential geometry). Vectors are the building blocks of Vector Field s and Vector Calculus . DEFINITIONS Informally, a vector is a quantity characterized by a Magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 Newton s". The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix ''R'', so that a coordinate vector x is transformed to x Accordingly, let, for example, each of two vectors be expressed as three space coordinates, and apply the formula for the cross product, resulting in three coordinates, which represent a third vector. If we rewrite the two vectors in rotated coordinates, and apply the formula for the cross product again, then the result is the original cross product in terms of rotated coordinates. Also, let, for example, a Vector Field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field. If we rewrite the original vector field in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values, and apply the formula for the curl based on these functions, then the result is the rewritten version of the original curl: also in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values. The same applies for dot product, gradient, divergence, vector addition and scalar multiplication. For these, also reflection in a plane can be applied. The scalars involved should not be transformed (e.g. in the case of a rotation by 180°, the scalar should not be multiplied by -1). Thus even in 1D we have to distinguish scalars and vectors: 2 × 3 = 6 can be interpreted as a scalar multiplication or a dot product, but not as a product of two vectors. Similarly differentiation in 1D can be interpreted as a gradient or a divergence: one of the two functions is scalar and one a vector, and the argument is a vector, ensuring invariance under inversion of the vectors without changing the scalars. Since rotation of the three Cartesian coordinate axes changes the formulas the same as an inverse rotation of the field itself, we can also conclude:
where rotation of a scalar field involves only rotation of the position vectors, while rotation of a vector field involves also a corresponding rotation of the vector field values. Note that the concept of corresponding rotations applies even if different coordinate systems are used for field values and position vectors, so that e.g. for one we multiply by an orthogonal matrix and for the other we add an angle to an angle coordinate. In order to use the usual formulas, e.g. to compute mechanical work, the ''x''-axis of forces should be in the same direction as the ''x''-axis of position, etc. When, as described above, coordinate rotations of position are accompanied by corresponding coordinate rotations of forces, this property is preserved. On the other hand, the origin of forces is simply at the zero force (no force), while the origin of position can be chosen as desired. For example, work depends on Displacement , which is the difference of positions and therefore does not depend on the origin. Position and function value of a vector field are often, but not necessarily, expressed in similar coordinate systems. For example gravitational field strength due to a particular point mass may be , with both the function value and the position vector in spherical coordinates. For the position vector the origin is chosen here at the center of the point mass; for the field strength the origin is simply at "zero field strength" anyway. How the other two coordinates are chosen does not matter in this case, because the field does not depend on them, and the field has no components in their directions. More generally, a vector is a Tensor of contravariant rank one. In differential geometry, the term ''vector'' usually refers to quantities that are closely related to tangent spaces of a differentiable Manifold (assumed to be three-dimensional and equipped with a Positive Definite Riemannian Metric ). (A Four-vector is a related concept when dealing with a 4 dimensional Spacetime Manifold in Relativity .) Examples of vectors include Displacement , Velocity , Electric Field , Momentum , Force , and Acceleration . Vectors can be contrasted with Scalar quantities such as Distance , Speed , Energy , Time , Temperature , Charge , Power , Work , and Mass , which have Magnitude , but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar. A related concept is that of a Pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under Improper Rotation s. Examples of pseudovectors include Magnetic Field , Torque , and Angular Momentum . (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying Symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a '''polar vector'''. See also Parity (physics) . Sometimes, one speaks informally of ''bound'' or ''fixed'' vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities. Examples in one dimension A force may be "15 N to the right", with coordinate 15 N if the basis vector is to the right, and −15 N if the basis vector is to the left. The magnitude of the vector is 15 N in both cases. A displacement may be "4 m to the right", with coordinate 4 m if the basis vector is to the right, and −4 m if the basis vector is to the left. The magnitude of the vector is 4 m in both cases. The work done by the force in the case of this displacement is 60 J in both cases. The force and displacement are vectors, the magnitudes are scalars, and the coordinates are neither. Generalizations In Mathematics , a vector is any element of a Vector Space over some Field . The spatial vectors of this article are a very special case of this general definition (they are ''not'' simply any element of ''' R '''''d'' in ''d'' dimensions), which includes a variety of mathematical objects ( Algebras , the Set of all Function s from a given Domain to a given linear Range , and Linear Transformation s). Note that under this definition, a Tensor is a special vector! REPRESENTATION OF A VECTOR Vectors are usually denoted in boldface, as a. Other conventions include or ''a'', especially in handwriting. Alternately, some use a Tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: Here the point ''A'' is called the ''tail'', ''base'', ''start'', or ''origin''; point ''B'' is called the ''head'', ''tip'', ''endpoint'', or ''destination''. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction. In the figure above, the arrow can also be written as or ''AB''. On a two-dimensional diagram, sometimes a vector Perpendicular to Plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre indicates a vector pointing out of the front of the diagram, towards the viewer. A circle with a cross inscribed in it indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip an Arrow front on and viewing the vanes of an arrow from the back. | ||
| :<math>\left\\mathbf{a} Ight\ | \sqrt{a_1^2+a_2^2+a_3^2}</math> |
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| The Length Of ''r'''''a''' Is ''r'''''a''' If The Scalar Is Negative, It Also Changes The Direction Of The Vector By 180<sup>o</sup> Two Examples (''r'' | -1 and ''r'' = 2) are given below: |
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| To Normalize A Vector '''a''' | ''a''<sub>2</sub>, ''a''<sub>3</sub> , scale the vector by the reciprocal of its length '''a''' That is: |
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