The dot product of two vectors a = [a1, a2, ... , an] and b = [b1, b2, ... , bn] is defined as:

$\Large \mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n$

where Σ denotes summation notation and n is the dimension of the vector space.

In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd. Similarly, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf. For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is

$\Large [1, 3, -5] \cdot [4, -2, -1] = (1 \times 4) + (3 \times -2) + (-5 \times -1) = 4 - 6 + 5 = 3$.

The dot product can also be obtained via transposition and matrix multiplication as follows:

$\Large \mathbf{b}^\mathrm{T}\mathbf{a} = \mathbf{a}^\mathrm{T}\mathbf{b}$,

where both vectors are interpreted as column vectors, and aT denotes the transpose of a, in other words the corresponding row vector.
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math_public
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Sat, 20 Aug 2011 11:06:30 GMT
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dirkjan
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Sat, 20 Aug 2011 11:06:30 GMT
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dirkjan
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dirkjan