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In Mathematics , the dot product, also known as the '''scalar product''', is an operation which takes two Vectors over the Real Numbers '''R''' and returns a real-valued Scalar quantity. It is the standard ''' Inner Product ''' of the Euclidean Space .


DEFINITION AND EXAMPLES

The dot product of two vectors (from an Orthonormal Vector Space ) a = ''a''2, … , ''a''''n'' and '''b''' = ''b''2, … , ''b''''n'' is by definition:
:\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 .

For example, the dot product of two three-dimensional vectors 3, −5 and −2, −1 is

:\begin{bmatrix}1&3&-5\end{bmatrix} \cdot \begin{bmatrix}4&-2&-1\end{bmatrix} = (1)(4) + (3)(-2) + (-5)(-1) = 3.

Using Matrix Multiplication and treating the (column) vectors as ''n''×1 Matrices , the dot product can also be written as:

:\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^T \mathbf{b} \,

where aT denotes the Transpose of the matrix a.

Using the example from above, this would result in a 1×3 matrix (i.e., vector) multiplied by a 3×1 vector (which, by virtue of the matrix multiplication, results in a 1×1 matrix, i.e., a scalar):

:\begin{bmatrix}
1&3&-5
\end{bmatrix}\begin{bmatrix}
4\-2\-1
\end{bmatrix} = \begin{bmatrix}
3
\end{bmatrix}.


GEOMETRIC INTERPRETATION


of a onto '''b''']]
In the Euclidean space there is a strong relationship between the dot product and Length s and Angle s. For a vector a, a•a is the square of its length, and, more generally, if '''b''' is another vector

  :'''a''' And '''b''' Denote The "http://wwwinformationdelightinfo/information/entry/length" class="copylinks">Length (magnitude) of '''a''' and '''b'''
  Since '''a'''cos(θ) Is The "http://wwwinformationdelightinfo/information/entry/scalar_resolute" class="copylinks">Scalar Projection of '''a''' onto '''b''', the dot product can be understood geometrically as the product of this projection with the length of '''b'''


  The "http://wwwinformationdelightinfo/information/entry/Inner_product_space" class="copylinks">Inner Product generalizes the dot product to Abstract Vector Spaces and is normally denoted by 〈'''a''', '''b'''〉 Due to the geometric interpretation of the dot product the Norm '''a''' of a vector '''a''' in such an Inner Product Space is defined as
  :<math>\\mathbf{a}\ \sqrt{\langle\mathbf{a}, \mathbf{a} angle},</math>




creating a triangle with sides ''a'', ''b'', and ''c''. According to the Law Of Cosines , we have
: c^2 = a^2 + b^2 - 2 ab \cos heta. \,
Substituting dot products for the squared lengths according to Lemma 1, we get
:
\mathbf{c} \cdot \mathbf{c}
= \mathbf{a} \cdot \mathbf{a}
+ \mathbf{b} \cdot \mathbf{b}
- 2 ab \cos heta. \,
                  ''(1)''
But as c ≡ '''a''' − '''b''', we also have
:
\mathbf{c} \cdot \mathbf{c}
= (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \,,
which, according to the Distributive Law , expands to
:
\mathbf{c} \cdot \mathbf{c}
= \mathbf{a} \cdot \mathbf{a}
+ \mathbf{b} \cdot \mathbf{b}
-2(\mathbf{a} \cdot \mathbf{b}). \,
                    ''(2)''
Merging the two c • c equations, ''(1)'' and ''(2)'', we obtain
:
\mathbf{a} \cdot \mathbf{a}
+ \mathbf{b} \cdot \mathbf{b}
-2(\mathbf{a} \cdot \mathbf{b})
= \mathbf{a} \cdot \mathbf{a}
+ \mathbf{b} \cdot \mathbf{b}
- 2 ab \cos heta. \,

Subtracting a • a + '''b''' • '''b''' from both sides and dividing by −2 leaves
: \mathbf{a} \cdot \mathbf{b} = ab \cos heta. \,
Q.E.D.


SEE ALSO




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