# Vector Inner-Multiplication Operation

(Redirected from Inner product)
• AKA: Canonical Dot Product, ·, Scalar Product.
• Context:
• input: two Vectors.
• output: a Non-Negative Real Number [0,infinity)
• It can (typically) be represented as $x·y$, or as $·(x,y)$.
• It can define a Dot Product Space.
• It can be used as a Vector Distance Function (such as kernel functions).
• It can be solved by a Vector Scalar Product.
• It can have properties of:
• $a \cdot b=0$, for $a=0$, or $b=0$.
• $a \cdot a = |a|^2$.
• $a \cdot b = b \cdot a$.
• $(a+b) \cdot c = a \cdot c + b \cdot c$ (Distributive law).
• $a \cdot b=|a||b|\cos\theta$, for $a \neq 0, b \neq 0$.
• $|a \cdot b| \leq |a||b|$ (Schwarz inequality).
• $a \cdot b=a_1b_1+a_2b_2+a_3b_3$, for vectors $a = a_1i+a_2j+a_3k$ and $b = b_1i+b_2j+b_3k$.
• It can be used to calculate the Vector Length Function by $\sqrt{\cdot(x,y)}$
• The real number $p=|a| \cos \theta$ (the vector component of $a$) in the direction of vector $b$ or the projection of vector $a$ in the direction of vector $b$. So how much the vector $a$ is projected in the direction of the vector $b$ can be found out by making dot product of the vector $a$ with unit vector of $b (= \frac{b}{|b|}=u)$. That is $a \cdot u=a \cdot \frac{b}{|b|}=|a|\frac{|b|}{|b|} \cos \alpha=|a| \cos \alpha= p$.
• $|p|$ is the length of the orthogonal projection of $a$ on a straight line $l$ in the direction of $b$.
• $p$ may be positive, zero or negative.

• Example(s):
• $(1,1) \cdot (1,1) = 1 \times 1 + 1 \times 1 = 2$, (notice that the distance is $\sqrt{2}$).
• $(1,0) \cdot (1,1) = 1 \times 0 + 1 \times 1 = 1$.
• $\cdot((1,0), (0,1)) = 1 \times 0 + 0 \times 1 = 0$, (two orthogonal vectors).
• $\cdot((1,2,3), (6,5,4)) = 28$.
• Counter-Example(s):
• See: Kernel Function, Hyperplane, Cosine Distance Metric, Vector Multiplication, Vector Length Function.