### Answer:

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 14:

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  =  ?
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