Dot product approaches zero as the magnitude of the vectors increase?

Question

Is there such thing? I'm doing some computations on mathematica and I noticed the dot product between two vectors are getting smaller and smaller as I increase the magnitude of the vectors, I'm not sure if there's something funny going on in my calculation or this is actually explained in vector calculus?

Answer 1

The dot product of two vectors $\textbf{A}$ and $\textbf{B}$ is given by

$$\textbf{A}.\textbf{B}=ABcos\theta$$

where $A$ and $B$ are the magnitudes of the vectors $\textbf{A}$ and $\textbf{B}$, respectively, and $\theta$ is the angle between the vectors $\textbf{A}$ and $\textbf{B}$.

In short, the dot product of $\textbf{A}$ with $\textbf{B}$ picks out the right component of $\textbf{B}$ in the direction of $\textbf{A}$.

Let

$$\textbf{A}=A_x\hat{x}+A_y\hat{y}+A_z\hat{z}$$ and

$$\textbf{B}=B_x\hat{x}+B_y\hat{y}+B_z\hat{z}$$

Then

$$\textbf{A}.\textbf{B}= A_xB_x+A_yB_y+A_zB_z=ABcos\theta$$

The dot product of two vectors is a scalar quantity. For the dot product of two non-zero vectors to be zero, the two vectors should be orthogonal (perpendicular in the Cartesian sense). I.e., $\textbf{A}$ and $\textbf{B}$ should be independent so that they share no common components with each other.

If you increase the magnitude of the vectors or the length of the vectors, then the corresponding components of each vector increase in magnitude. Since you have only increased the magnitude, the angle between them, $\theta$, will not vary. So the dot product should increase. If you change the vectors so that the angle between them also decrease, then the dot product decreases.

Answer 2

Recall that the dot product will be zero for orthogonal vectors. If your larger size is also changing the effective angle, this could account for the unexpected result.

Answer 3

A dot product is defined as the product of two vector amplitudes times the cosine of the angle between them. In one dimension, that 'cosine' term is either $1$ or $-1$. The RMS average is $1$. In two dimensions, it has an RMS average value of $1/\sqrt{2}$.

This generalizes, and between two $N$-dimensional vectors, the RMS average value of the cosine is $1/\sqrt{N}$.

Thus, dot products of vectors of higher dimensionality have a factor that decreases with $\sqrt{N}$ unless their orientations are non-random. The $1/\sqrt{N}$ factor goes with the number of orthogonal components that span the vector space, though, not with the amplitude of the vectors in question.