Vector
A vector is a magnitude with a direction. Much of physics deals with vectors. An -dimensional vector can be thought of as an ordered
-tuple of numbers within angle brackets. The set of vectors in some space is an example of a vector space.
Contents
Description
Every vector has a starting point
and an endpoint
. 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
(i)
(ii)
(iii)
(iv)
...
Vector Operations
Dot (Scalar) Product
Consider two vectors and $\bold{v}=\langlev_1, v_2,\ldots,v_n\rangle$ (Error compiling LaTeX. Unknown error_msg) in
. The dot product is defined as
.
Cross (Vector) Product
The cross product between two vectors and
in
is defined as the vector whose length is equal to the area of the parallelogram spanned by
and
and whose direction in accordance with the right-hand rule.
Triple Scalar product The triple scalar product of three vectors is defined as
. Geometrically, the triple scalar product gives the signed area of the parallelpiped determined by
and
. It follows that
![$(\bold{a}\times\bold{b})\cdot \bold{c} = (\bold{c}\times\bold{a})\cdot \bold{b} = (\bold{b}\times\bold{c})\cdot \bold{a}.$](http://latex.artofproblemsolving.com/2/f/1/2f1162b7eb4c301897dba471b3f4372b0a70cec7.png)
It can also be shown that
![$(\bold{a}\times\bold{b})\cdot \bold{c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}.$](http://latex.artofproblemsolving.com/f/a/6/fa6e7cbf53bb81ffed2d33a8e63a2942274b9def.png)
Triple Vector Product
See Also
Related threads from AoPS forum
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