A good answer might be:

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.)


Dot Product in Three Dimensions

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   =   ( a1, a2, a3 )T
Let b   =   ( b1, b2, b3 )T

Then the dot product is:

a · b   =   a1b1 + a2b2 + a3b3

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

QUESTION 13:

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  =  ?