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In Mathematics , the dot product, also known as the '''scalar product''', is a Binary Operation which takes two Vectors and returns a Scalar quantity. It is the standard ''' Inner Product ''' of the Euclidean Space .

The dot product of two vectors a = ''a''2, … , ''a''''n'' and '''b''' = ''b''2, … , ''b''''n'' is by definition

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

where Σ denotes Summation Notation . For example, the dot product of two three-dimensional vectors 3, −2 and −2, −1 is

: 3, −2 · −2, −1 = 1(4) + 3(−2) + −2(−1)= 0

Using Matrix Multiplication and treating the Row Vector s as 1×''n'' Matrices , the dot product can also be written as:

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

where bT denotes the Transpose of the matrix b. 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&-2\end{bmatrix}\begin{bmatrix}4\-2\-1\end{bmatrix} = \begin{bmatrix}0\end{bmatrix}


GEOMETRIC INTERPRETATION


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 if '''b''' is another vector

: \mathbf{a} \cdot \mathbf{b} = a \, b \cos heta \;

where ''a'' and ''b'' denote the Length of a and '''b''', and θ is the Angle between them.

Since ''a''·cos(θ) is the Projection of a onto '''b''', the dot product can be understood geometrically as the product of this projection with the length of '''b'''.

As the Cosine of 90° is zero, the dot product of two Perpendicular vectors is always zero. If a and '''b''' have length one (they are Unit Vector s), the dot product simply gives the cosine of the angle between them. Thus, given two vectors, the angle between them can be found by rearranging the above formula:

  The "http://wwwinformationdelightinfo/encyclopedia/entry/Inner_product_space" class="copylinks">Inner Product generalizes the dot product to Abstract Vector Spaces , it 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>,