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Wednesday, July 22, 2009

Vector Algebra 1

 Here in this post we will go through a quick recap of vector algebra keeping in mind that reader already had detail knowledge and problem solving skills related to the topic being discussed. Here we are briefing Vector Algebra because concepts of electrostatics , electromagnetism and many more physical phenomenon can best be conveniently expressed using this tool.

A vector is a quantity that requires both a magnitude (= 0) and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as
A = Axi+ Ayj+Azk

Two vectors A and B can be added together to give another resultant vector C.
C = A + B

Two vectors A and B can be subtracted to give another resultant vector D.
D = A - B = A + (-B)

When we multiply any vector A with any scalar quantity 'n' then it's direction remains unchanged and magnitude gets multiplied by 'n'. Thus,
n(A) = nA
Scalar multiplication of vectors is distributive i.e.,
n(A + B) = nA +nB

Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as A.B thus,
A.B = |A| |B| cosθ
where θ is the angle between two vectors.
Result of dot product of two vectors is a scalar quantity.
Dot product is commutative : A.B = B.A
Dot product is distributive : A . (B+C) = A.B + A.C also A.A = |A|2

Cross product or vector product of two vectors A and B is defined as
A x B = |A| |B| sinθ nˆ
where nˆ is the unit vector pointing in the direction perpandicular to the plane of both A and B. Result of vector product is also a vector quantity.
Cross product is distributive i.e., A x (B + C) = (A x B) + (A x C) but not commutative and the cross product of two parallel vectors is zero.

In our next post we'll study vector algebra in component form and also lern about vector triple products.

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