Vector

Revision as of 16:08, 11 August 2006 by Aryth (talk | contribs) (Vector Operations)

A vector is a magnitude with a direction. Much of physics deals with vectors. An $\displaystyle n$-dimensional vector can be thought of as an ordered $\displaystyle n$-tuple of numbers within angle brackets. The set of vectors in some space is an example of a vector space.


Description

Every vector $\vec{PQ}$has a starting point $P<x_1, y_1>$ and an endpoint $Q<x_2, y_2>$. Since the only thing that distinguishes one vector from another is its magnitude,i.e. length, and direction, vectors can be freely translated about a plane without changing them. Hence, it is convenient to consider a vector as originating from the origin. This way, two vectors can be compared only by looking at their endpoints. The magnitude of a vector, denoted is found simply by using the distance formula.

Properties of Vectors

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(ii)

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Vector Operations

Dot (Scalar) Product (proof as well? ) (--use law of cosines; I'm not good at proofs--from Aryth)

Consider two vectors $\bold{u}=<u_1,u_2,...,u_n>$ and $\bold{v}=<v_1, v_2,...,v_n>$. The dot product is defined as $\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+...+u_nv_n$. In two or three dimensions, the dot product has the special geometric property that $\cos{\theta}=\frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|\|\bold{v}\|}$


Cross (Vector) Product


Triple Scalar product


Triple Vector Product

See Also

Related threads from AoPS forum


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