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Introduction to Vectors:

Vectors: Vectors are the quantities which are have both a magnitude and a direction. Vectors only impart magnitude and direction.

Consider the figure below.

Introduction to Vectors

Each line segment in the above figure represents the same vector and each vector starts at specific point and move 2 unit left and 5 units up. The notation for the vector is v-> =(-2,5)

The vector denotes a magnitude and a direction of a quantity while the point denotes a location in space. A representation of the vector v-> = (a1,a2) in two dimensional space is any directed line segment, directed line segment from the point A=(x,y) to the point B= (x+ a1,y+ a2). Likewise a representation of the vector v-> = (a1,a2,a3) in three dimensional space is any directed line segment, directed line segment, from the point A=(x,y,z) to the point B= (x + a1,y+ a2+ z+ a3).

The magnitude, or length, of the vector v-> =(a1,a2,a3) is given by,

║v->║=√(a21 + a22 + a23 )

Unit vector: Any vector with magnitude of 1, i.e ║ u-> ║ =1 , is called a unit vector.

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Zero Vector: The vector w ⃗ =(0,0) that we saw in the first example is called a zero vector since its components are all zero. Zero vectors are often denoted by 0 ⃗. Be careful to distinguish 0 (the number) from 0 ⃗ (the vector). The number 0 denotes the origin in space, while the vector 0 ⃗ denotes a vector that has no magnitude or direction.

Standard Basis Vectors:

The fourth vector from the second example, i ⃗ =(1,0,0), is called a standard basis vector. In three dimensional space there are three standard basis vectors,

i ⃗=(1,0,0)
j ⃗=(0,1,0)
k ⃗=(0,0,1)

In two dimensional spaces there are two standard basis vectors,

i ⃗=(1,0)
j ⃗=(0,1)

This standard basis vectors are also unit vectors.

Properties of vector:

If v ⃗ , w ⃗ and u ⃗ are vectors (each with the same number of components) and a and b are two numbers then we have the following properties.

v ⃗ + ( w) ⃗ = w ⃗ + v ⃗
u ⃗ + (v ⃗ + ( w) ⃗) = (u ⃗ + ( v) ⃗) + w ⃗
v ⃗ + 0 ⃗ = v ⃗
1v ⃗ = v ⃗
a ( v ⃗ + w ⃗ ) = av ⃗ +aw ⃗
(a + b) v ⃗ = av ⃗ +bv ⃗