Vector algebra is collection of n scalars that is real numbers.Lower case letters are used to represent the vectors.In n-dimensional space, scalar in point or arrow.vector algebra has some properties.They are coordinate free,dimension independent,no position and use the basic operations like addition,subtraction and position vectors.Scalar product and vector product in two ways.
Addition and subtraction in vector algebra and its geometry:
Equal dimensions are used to add or subtract the two vectrors that is one element is add or subtract by another element.In vector algebra,parallelogram rule is used.
In equation, 1). (a+b)=(a1+b1, a2+b2 ,.........an+bn).
2). (a-b)=(a1-b1, a2-b2 ,.........an-bn).
The above geometry is example for addition operation.Here two vectors are represented in both sides that is right and left side.
Vectors- algebra and its geometry:
Scalar produt give the two equal length vectors as input and get the one number output.Inner product is principle of vector algebra.A geometry define the entire operation in detail.
Consider two vectors `veca` and `vecb` .These two vectors are non zero vectors at an angle Θ. Dot product of vectors is `veca` . `vecb` and defined as | `veca` | |`vecb` | cosΘ.
`veca` .`vecb` = | `veca` | |`vecb` | cosΘ = `veca` `vecb` cosΘ
Equal dimensions are used in vectors.Cross product is different with dot product.Scalar product has different properties.The above geometry illustrate product of two vectors of a and b.
Property 1: Scalar product of two vectors is commutative.
`veca` .`vecb` =`vecb` .`veca`
Property 2: i). When `veca` and `vecb` are collinear and in same direction.So Θ=0.
`veca` .`vecb` = | `veca` | |`vecb` | cosΘ = | `veca` | |`vecb` | (1) =ab.
ii).when `veca`and `vecb` are collinear and in opposite direction.So Θ =Π.
`veca` .`vecb` = | `veca` | |`vecb` | (-1) = -ab.
Property 3: Vector length = |a|=`sqrt(a.a)`
Property 4: Two vectors angle a.b / |a||b|=cos Θ.
Property 5: when vectors are perpendicular, dot product is 0.