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Scalar multiplication of a vector by a factor of 3 stretches the vector out. A vector may also be multiplied, or re-scaled, by any real number r. In the context of conventional vector algebra, these real numbers are often called scalars (from scale) to distinguish them from vectors.
The magnitude of a vector (such as distance) is another example of an invariant, because it remains fixed even if geometrical vector components vary. (For example, for a position vector of length meters, if all Cartesian basis vectors are changed from meters in length to meters in length, the length of the position vector remains unchanged at ...
A numeric example with three equations and two unknowns: In case there are more equations than variables and the equations have a solution, then each of the k-vector quotients will be scalars. To illustrate here is the solution of a simple example with three equations and two unknowns.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus , an extension of GA that incorporates differentiation and integration , can be used to formulate other theories such as complex analysis and differential ...
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used.
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
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