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

Consider the figure below.

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, 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, , 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^{->}║=√(a^{2}_{1} + a^{2}_{2} + a^{2}_{3} )

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

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.

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.

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 ⃗

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