Let u, v, w ∈ R^3
 Show that:
The vector u − projvu is orthogonal to v

To show that the vector \( u - \text{proj}_v u \) is orthogonal to \( v \), let's break it down step by step.

### Step 1: Projection formula
The projection of a vector \( u \) onto a vector \( v \), denoted as \( \text{proj}_v u \), 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 gives \( |v|^2 \), the square of the magnitude of \( v \)).

### Step 2: The vector \( u - \text{proj}_v u \)
We want to show that the vector \( u - \text{proj}_v u \) is orthogonal to \( v \). For two vectors to be orthogonal, their dot product must be zero. Thus, we need to show that:
\[
\left( u - \text{proj}_v u \right) \cdot v = 0
\]

### Step 3: Expand the expression
Using the definition of \( \text{proj}_v u \), we expand the dot product:
\[
\left( u - \frac{u \cdot v}{v \cdot v} v \right) \cdot v
\]
Distribute the dot product over the subtraction:
\[
u \cdot v - \left( \frac{u \cdot v}{v \cdot v} v \right) \cdot v
\]

### Step 4: Simplify the second term
Now, simplify the second term. We know that \( v \cdot v = |v|^2 \), so:
\[
\left( \frac{u \cdot v}{v \cdot v} v \right) \cdot v = \frac{u \cdot v}{v \cdot v} (v \cdot v) = \frac{u \cdot v}{v \cdot v} \cdot v \cdot v = u \cdot v
\]

### Step 5: Final expression
Now substitute this result back into the expanded expression:
\[
u \cdot v - u \cdot v = 0
\]

Thus, \( \left( u - \text{proj}_v u \right) \cdot v = 0 \), which means that \( u - \text{proj}_v u \) is orthogonal to \( v \).

### Conclusion
We have shown that the vector \( u - \text{proj}_v u \) is orthogonal to \( v \), as required.

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Would you like to explore more details on any part of this explanation? Here are 5 related questions that can deepen your understanding:

1. How does the dot product relate to the concept of orthogonality in \( \mathbb{R}^3 \)?
2. What are the geometric interpretations of the projection of one vector onto another?
3. Can you show a similar proof for higher dimensions, such as \( \mathbb{R}^n \)?
4. How would this result apply if the vectors were not in \( \mathbb{R}^3 \) but in a different vector space?
5. How does this proof change if \( v \) is the zero vector?

**Tip:** Orthogonal vectors always have a dot product of zero, which is key to checking for perpendicularity in vector spaces.

Let u, v, w ∈ R^3. Show that the vector u − proj_v(u) is orthogonal to v.

Explore how to prove that the vector u − proj_v(u) is orthogonal to v in R^3. This solution covers vector projection, orthogonality, and key linear algebra concepts, with step-by-step reasoning. Ideal for undergraduate students learning about vector spaces and projections.

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