Projection of Vectors

Standard Definition: The projection of vector $\mathbf{A}$ onto another vector $\mathbf{B}$ is the component of $\mathbf{A}$ that lies in the direction of $\mathbf{B}$. It's found by taking the dot product of $\mathbf{A}$ with the unit vector in the direction of $\mathbf{B}$. Mathematically, the projection of $\mathbf{A}$ onto $\mathbf{B}$ is given by:

Where:

• $\mathbf{A} \cdot \mathbf{B}$ is the dot product of the vectors.
• $|\mathbf{B}|$ is the magnitude (or norm) of vector $\mathbf{B}$.

ELI5 with Real-life Example: Imagine you're standing outside on a sunny day, and you see your shadow on the ground. Your shadow is essentially a "projection" of yourself onto the ground. The sun's position determines the direction and length of the shadow. If the sun is right above you, your shadow is the shortest (directly beneath you). As the sun sets, your shadow becomes longer and changes direction.

In this analogy:

• You are like the vector $\mathbf{A}$.
• The ground represents the direction of vector $\mathbf{B}$.
• Your shadow is the projection of $\mathbf{A}$ onto $\mathbf{B}$.

The way the sunlight shines on you and casts your shadow onto the ground is similar to how one vector is projected onto another in linear algebra. The projection tells us how much of one vector lies in the direction of another vector.

Visualization

Here's the visualization illustrating the concept of vector projection:

• The blue arrow represents "you" (vector $A$).
• The green arrow represents the "ground direction" (vector $B$).
• The red dotted arrow represents "your shadow," which is the projection of $A$ onto $B$.

The dashed line connecting the tip of the blue arrow to the red arrow shows the orthogonal (right-angle) relationship between the original vector and its projection onto the other vector.

This visualization is akin to how your shadow is cast on the ground, with the shadow's length and direction determined by the sun's position and the ground's orientation.

Let's use a practical example involving navigation and sailing:

Example

Imagine you're sailing on a boat, and you need to get to an island that's directly north of your starting point. However, due to some constraints (maybe wind patterns or water currents), you end up sailing in a slightly northeast direction. After a while, you want to know how much distance you've actually covered towards your northward target. This is a perfect situation to use vector projection.

Setting up the Vectors

1. Desired Direction (North): Let's represent this as vector $\mathbf{N}$. For simplicity, let's assume you wanted to sail straight north for 10 units (this could be kilometers, miles, etc.). $\mathbf{N} = \begin{bmatrix} 0 \\ 10 \end{bmatrix}$

2. Actual Path (Northeast): Let's assume you sailed northeast for a distance of 10 units. This means you've covered equal distances north and east. Let's represent this path as vector $\mathbf{A}$. $\mathbf{A} = \begin{bmatrix} 7 \\ 7 \end{bmatrix}$ (This is an approximation for simplicity.)

Let's perform these calculations

Calculations

1. Dot Product of $\mathbf{A}$ and $\mathbf{N}$: ( 70 )
2. Norm (Magnitude) of the Desired Direction $\mathbf{N}$: ( 10 ) units
3. Projection of $\mathbf{A}$ onto $\mathbf{N}$: $\text{proj}_{\mathbf{N}} \mathbf{A} = \begin{bmatrix} 0 \\ 7 \end{bmatrix}$
4. Distance covered in the desired (northward) direction: ( 7 ) units

Interpretation

Despite sailing in a northeast direction for 10 units, you've effectively covered 7 units in your desired northward direction. The remaining 3 units took you eastward, away from your straight path to the island.

This example illustrates how vector projection can help determine how much of a vector's magnitude is in the direction of another vector. In real-life scenarios like navigation, understanding this can be crucial for course corrections and efficient travel.