# Index Notation as Projection Operator and Division of Vectors

I came across a nice equivalence between the standard index notation for vectors and the vector projection operator. This works for vectors of any dimension, but I’ll explain it for vectors in three dimensions.
A vector a can be expressed by its coordinates in a coordinate system:

(ax, ay, az)

These are three numbers that depend on the chosen coordinate system.  Any coordinate system can be described by three vectors x, y and z.  Given these three vectors, we can then write the components of a as

ax = (a · x) / ‖x‖².

We can interpret this as a binary operation with arguments a and x. This operator corresponds to the projection of a onto x, expressed in units where x has length one:

Using the points laid out as in this graphic, we can write the operator as

This sign of ax is given by the relative directions of AC and AD.

Since this operator only depends on the dot product (a · x) and the norm ‖x‖, its value is independent of the coordinate system in which it is computed.

As a binary operator, ax is linear in the first argument, i.e., for any number c we have

(ca)x = c (ax),

and for any two vectors a and b we have

(a + b)x = ax + bx.

The operator is inverse-linear in the second operator:

a(cx) = c−1 (ax)

The operator is defined for all vectors a and x, except when x is the zero vector.

When the two vectors a and x have the same norm ‖a‖ = ‖x‖, then exchanging the two operands does not change the value:

ax = xa  ⇔  ‖a‖ = ‖x

This value ax = xa then equals the cosine of the angle between a and x.  In the general case, the operator is however not commutative.

The operator is zero if and only if the two vectors are orthogonal:

ax = 0  ⇔  ax

These properties are similar to those of scalar division, although this operation is of course not a proper division, since it returns not a vector, and cannot be expressed in terms of products and inverses. Note also how by coincidence, the notation ax puts a in a high position and x in a low position, just as in a fraction.