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

|ax| = AC / AD

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.

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