This *does* make sense:

2( -1, 2)^{T}·( 4, 1 )^{T}= ( -2, 4)^{T}·( 4, 1 )^{T}= -2*4 + 4*1 = -8 + 4 = -4

(Notice that there is no "dot" between the 2 and the vector following it, so this means "scaling," not dot product.)

The dot product is defined for 3D column matrices.
The idea is the same: *multiply corresponding elements of both column matrices,
then add up all the products*.

- Let
**a**= ( a_{1}, a_{2}, a_{3})^{T} - Let
**b**= ( b_{1}, b_{2}, b_{3})^{T}

Then the dot product is:

a · b= a_{1}b_{1}+ a_{2}b_{2}+ a_{3}b_{3}

Both column matrices must have the same number of elements.

- (1, 2, 3)
^{T}**·**(6, 7, 8)^{T}= 1*6 + 2*7 + 3*8 = 44 - ( -1, 2, -3)
^{T}**·**(1, -2, 3)^{T}= (-1)(1) + (2)(-2) + (-3)(3) = -1 + -4 + -9 = -14

Nothing wrong with having variables as elements of the vectors:

- (1, 2, 3)
^{T}**·**(x, y, z)^{T}= x + 2y + 3z

You must be itching to try this yourself (or is that your allergy to math acting up again?)

( 4, 0, -3)^{T}·(0, -2, 0)^{T}= ?