Dot product

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Dot product as correlation

Here’s how the dot product acts like a measure of “correlation” or alignment between two vectors $\mathbf{v}$ and $\mathbf{w}$. Recall

• If the result is large positive, the two vectors tend to point in the same direction (positively correlated).
• If it’s large negative, they point in opposite directions (negatively correlated).
• If it’s zero, they’re orthogonal—no linear alignment (uncorrelated).

To illustrate, we’ll use

and three choices of ${\mathbf{v}}_2$:

1. Positive Correlation
   ${\mathbf{v}}_2=[-3,-2,-1,0,1,2,3]$
   ${\mathbf{v}}_1\cdot {\mathbf{v}}_2=28$
   points roughly along the same “trend.”

2. Negative Correlation
   ${\mathbf{v}}_2=[3,2,1,0,-1,-2,-3]$
   ${\mathbf{v}}_1\cdot {\mathbf{v}}_2=-28$
   one increases while the other decreases.

3. Zero Correlation
   ${\mathbf{v}}_2=[1,1,1,1,1,1,1]$
   ${\mathbf{v}}_1\cdot {\mathbf{v}}_2=0$
   balanced ups and downs cancel out—no net alignment.