Dot Product

Standard Definition: The dot product (or scalar product) of two vectors $\mathbf{A}$ and $\mathbf{B}$ is a scalar value obtained by multiplying the magnitudes of the vectors by the cosine of the angle $\theta$ between them. Mathematically:

Alternatively, for vectors in Cartesian coordinates:

(where $A_x, A_y, A_z$ and $B_x, B_y, B_z$ are the components of vectors $\mathbf{A}$ and $\mathbf{B}$ respectively.)

ELI5 with Real-life Example: Imagine you and a friend are pushing a broken car on a straight road. Both of you are pushing in slightly different directions. The dot product helps determine how much of your combined effort is actually helping to move the car forward on the road.

Say you're pushing the car with a force equivalent to carrying 5 heavy grocery bags, and your friend with a force equivalent to 4 grocery bags. However, your friend is pushing slightly to the side. The dot product would calculate the "effective" force from both of you that contributes directly to moving the car straight down the road. If both of you were pushing in the exact same direction, the dot product would simply be like adding up all the grocery bags' effort. But since the directions are slightly different, the dot product accounts for that and gives the combined "useful" effort.

In many practical scenarios, like physics and engineering, the dot product is used to find the projection of one vector onto another or to check the alignment of two vectors.

Visualization

Here's a visualization illustrating the dot product using the lawn mower example:

• The blue arrow represents the force direction you're applying on the lawn mower (vector ( A )).
• The green arrow represents the actual direction in which the lawn mower is designed to move (vector ( B )).
• The dotted red arrow represents the "effective" force resulting from your push in the direction the mower moves. This is the projection of your force onto the mower's direction, and it's what actually contributes to moving the mower forward.

The dot product essentially measures how much of your force (blue arrow) aligns with the direction of the mower (green arrow). The more aligned your force is with the mower's direction, the longer the dotted red arrow will be, indicating a more effective push.

In this visualization, even though you're pushing somewhat diagonally, only the portion of your force that aligns with the mower's direction (the dotted red arrow) is useful for moving the mower forward. The rest is "wasted" pushing sideways.

Real World Examples

The dot product is a fundamental operation in vector mathematics, and its applications touch various aspects of our daily lives. Here are some real-life examples:

1. GPS Navigation:

   • When your GPS calculates the quickest route, it's considering the "direction" you're headed in and the "direction" of your desired destination. The dot product helps in determining how "aligned" these directions are and suggests turns or route changes accordingly.

2. Light and Surfaces:

   • Think about when sunlight hits a window. The amount of light that gets reflected versus the amount that passes through can be thought of in terms of the dot product. The angle of sunlight and the window's orientation play a role in determining the brightness of the light inside the room.

3. Work in Physics:

   • If you've ever pushed a lawn mower, the "work" you do (in physics terms) is the dot product of the force you apply and the direction the mower moves. If you push straight ahead, it's efficient. If you push sideways, not much work gets done in moving the mower forward.

4. Audio and Acoustics:

   • When setting up stereo speakers, the "stereo image" (how you perceive the location of sounds) can be influenced by the angle between the speakers and your ears. The dot product can help in determining the optimal angle for speaker placement relative to the listener for the best sound experience.

5. Computer Graphics:

   • When you play video games or use applications with 3D graphics, the way light interacts with objects (shading, reflections) is often calculated using dot products. It helps in determining how light sources affect the appearance of surfaces in the virtual environment.

6. Economics and Market Analysis:

   • When analyzing market trends, analysts might consider the "alignment" or "correlation" between different product sales. If two products have sales trends that move in similar directions, the dot product of their sales vectors might be used to quantify this alignment.

Each of these examples involves some notion of alignment or projection, concepts that the dot product excels at quantifying.