Vector Algebra

Vector Algebra

A scalar is a value that can be represented by a single number.
A vector is also quantity that has both magnitude (size) and direction. Vectors are often written in bold type, to distinguish them from scalars. Velocity of a moving point is an example of a vector quantity.In two dimensional space a vector may be represented by two scalar components, in three dimensions a vector may be represented by three scalar components. Most simply these are Cartesian coordinates.  However in 2D vectors can be written in polar coordinates and in 3D they can be written in spherical or cylindrical coordinates.

Elementary Vector Algebra Vector Arithmetic Notes
Vectors follow obvious rules of addition and subtraction. Also the multiplication of a vector by a scalar is straightforward. For example if a = (a₁,a₂,a₃),b = (b₁,b₂,b₃) are vectors and a is a scalar then

a + b = (a1,a2,a3)+(b1,b2,b3) = (a1+b1,a2+b2,a3+b3)

a - b = (a1,a2,a3)-(b1,b2,b3) = (a1-b1,a2-b2,a3-b3)

aa = (aa1,aa2,aa3)

Dot Product
The dot product of two vectors a and b is defined as follows:

a . b = (a1,a2,a3).(b1,b2,b3) = a1 b1+ a2 b2+ a3 b3

Cross Product
The cross product of two vectors a and b is defined as follows:

a ×b = (a1,a2,a3) ×(b1,b2,b3) = (a2 b3 - a3 b2, a3 b1 - a1 b3, a1 b2 -a2 b1)