Here's my interpretation of dot notation:

a = [2,6]
b = [1,4]
c = [0,8]
a . b . c = (2*6)+(1*4)+(0*8) = 12 + 4 + 0 = 16

What is the significance of 16? Apparently it's a scalar.

Am I right in thinking that a scalar is the number we times a unit vector by to get a vector that has a scaled up magnitude but the same direction as the unit vector? So again, what is the relevance of 16? When is it used? It's not the magnitude of all the vectors added up.

The magnitude of all of them is calculated as follows:

sqrt( ax * ax + ay * ay ) + sqrt( bx * bx + by * by ) + sqrt( cx * cx + cy * cy)
sqrt( 2 * 2 + 6 * 6 ) + sqrt( 1 * 1 + 4 * 4 ) + sqrt( 0 * 0 + 8 * 8)
sqrt( 4 + 36 ) + sqrt( 1 + 16 ) + sqrt( 0 + 64)
sqrt( 40 ) + sqrt( 17 ) + sqrt( 64)
6.3 + 4.1 + 8
10.4 + 8
18.4

So I don't really get this diagram:

Attempting with sensible numbers:

a = [1,0]
b = [4,3]

a . b = (1*0) + (4*3) = 0 + 12 = 12

So what exactly is a . b describing here? The magnitude of that vector? Because that isn't right:

the 'a.b' vector = [4,0]
sqrt( x*x + y*y )
sqrt( 4*4 + 0*0 )
sqrt( 16 + 0 )
4

So what is 12 describing?