在学习SVM时，遇到了dot product的问题，一时忘了在algebra下定义的向量内积和在geometry下定义的向量内积为何相等，查找了一下资料，发现很有趣，故记录如下。
This operation can be defined either algebraically or geometrically.
- Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers.
- Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.
Algebraic definition
The dot product of two vectors A = [A1, A2, …, An] and B = [B1, B2, …, Bn] is defined as:
Geometric definition
In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction that the arrow points. The magnitude of a vector A is denoted by $|\mathbf{A}|$.
The dot product of two Euclidean vectors A and B is defined by
where θ is the angle between A and B.
Equivalence of the definitions
If e1,…,en are the standard basis vectors in Rn, then we may write
The vectors $e_i$ are an orthonormal basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length
$\mathbf e_i\cdot\mathbf e_i=1$
and since they form right angles with each other, if $i ≠ j$,
$\mathbf e_i\cdot\mathbf e_j = 0.$
Also, by the geometric definition, for any vector ei and a vector A, we note
$\mathbf A\cdot\mathbf e_i = |\mathbf A|\,|\mathbf e_i|\cos\theta = |\mathbf A|\cos\theta = A_i$,
where Ai is the component of vector A in the direction of ei.
Now applying the distributivity of the geometric version of the dot product gives
which is precisely the algebraic definition of the dot product. So the (geometric) dot product equals the (algebraic) dot product.