Standard Basis Vectors Revisited

In the preceding section we introduced the idea of standard basis vectors with no really discussing why they were significant.  We can now do that.  Let us start with the vector 

a^→ = (a₁, a₂, a₃)

We can make use of the addition of vectors to break this up as follows,

a^→ = (a₁, a₂, a₃)

= (a₁, 0, 0) + (0, a₂, 0) + (0, 0, a₃)

By using scalar multiplication we can further rewrite the vector like,

a^→ = (a₁, 0, 0) + (0, a₂, 0) + (0, 0, a₃)

= a₁ (1,0,0) + a₂ (0, 1, 0) + a₃ (0,0,1)

At last, notice that these three new vectors are just the three standard basis vectors for three dimensional space.

(a₁, a₂, a₃) = a₁i^→ + a₂j^→ +a₃k^→

Thus, we can take any vector and write out it in terms of the standard basis vectors.  From this point on we will make use of the two notations interchangeably so ensure that you can deal with both notations.