Dot Product¶
Formula¶
\[
x\cdot y = \sum_{i=1}^n x_i y_i
\]
Parameters¶
- \(x,y\in\mathbb{R}^n\): vectors
What it means¶
Measures alignment between vectors; equals \(\|x\|\,\|y\|\cos\theta\).
What it's used for¶
- Measuring similarity and angles between vectors.
- Projection and least squares derivations.
Key properties¶
- Bilinear and symmetric
- Defines length: \(\|x\|_2 = \sqrt{x\cdot x}\)
Common gotchas¶
- Dot product depends on coordinate system only through the chosen basis.
- In complex spaces, use conjugate transpose (Hermitian inner product).
Example¶
If \(a=(1,2)\) and \(b=(3,4)\), then \(a\cdot b=11\).
How to Compute (Pseudocode)¶
Input: vectors x[1..n], y[1..n]
Output: dot product x . y
s <- 0
for i from 1 to n:
s <- s + x[i] * y[i]
return s
Complexity¶
- Time: \(O(n)\)
- Space: \(O(1)\) extra space
- Assumptions: Vectors have length \(n\); complex-valued vectors use a conjugate inner product convention