To demonstrate that the projection of $u$ onto the projection of $u$ onto $v$ is equal to the projection of $u$ onto $v$, let's break it down step-by-step mathematically.

Step 1: Definition of projection

Recall that the projection of a vector $u$ onto a vector $v$ is given by the formula:

$\text{proj}_{v}(u) = \frac{u \cdot v}{v \cdot v} v$

where $u \cdot v$ is the dot product of $u$ and $v$, and $v \cdot v$ is the dot product of $v$ with itself, which is $|v|^2$, the square of the magnitude of $v$.

Step 2: Projection of $u$ onto $v$

Let’s denote the projection of $u$ onto $v$ by $p_1$. From the projection formula, we have:

$p_1 = \text{proj}_v(u) = \frac{u \cdot v}{v \cdot v} v$

Step 3: Projection of $p_1$ onto $v$

Next, we calculate the projection of $p_1$ onto $v$. Using the formula for projection again:

$\text{proj}_v(p_1) = \frac{p_1 \cdot v}{v \cdot v} v$

Substitute $p_1 = \frac{u \cdot v}{v \cdot v} v$ into this expression:

$\text{proj}_v(p_1) = \frac{\left( \frac{u \cdot v}{v \cdot v} v \right) \cdot v}{v \cdot v} v$

Now, simplify the dot product $\left( \frac{u \cdot v}{v \cdot v} v \right) \cdot v$:

$\left( \frac{u \cdot v}{v \cdot v} v \right) \cdot v = \frac{u \cdot v}{v \cdot v} (v \cdot v) = u \cdot v$

So we get:

$\text{proj}_v(p_1) = \frac{u \cdot v}{v \cdot v} v$

Step 4: Comparing with $\text{proj}_v(u)$

Notice that the result we obtained for $\text{proj}_v(p_1)$ is exactly the same as the formula for $\text{proj}_v(u)$. Therefore, we have shown that:

$\text{proj}_v(p_1) = \text{proj}_v(u)$

Conclusion:

Thus, the projection of $u$ onto the projection of $u$ onto $v$ is equal to the projection of $u$ onto $v$:

$\text{proj}_v(\text{proj}_v(u)) = \text{proj}_v(u)$

This completes the demonstration.

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Would you like any further clarifications or a deeper explanation on projections?

Here are some related questions for further exploration:

1. What does the projection of a vector onto another vector geometrically represent?
2. How does the projection of a vector relate to the concept of orthogonality?
3. What is the relationship between the projection of a vector and its component along another vector?
4. How does the projection formula change when working in higher dimensions (e.g., 3D space)?
5. Can you derive the formula for projection when using the cross product instead of the dot product?

Tip: Projections are useful for finding components of vectors along certain directions, and this property of projections being idempotent (i.e., projecting twice yields the same result) is very helpful when analyzing vector spaces.