If x, y ∈ Rn
, then x and y are called perpendicular (or orthogonal) if hx, yi = 0. If x and y are
perpendicular, prove that |x + y|
2 = |x|
2 + |y|
2
.

Let's prove that if \( x \) and \( y \) are perpendicular vectors, then \( |x + y|^2 = |x|^2 + |y|^2 \).

### Given:
- \( x \) and \( y \) are vectors in \( \mathbb{R}^n \).
- \( x \) and \( y \) are perpendicular (or orthogonal), meaning \( \langle x, y \rangle = 0 \).

### To Prove:
\[ |x + y|^2 = |x|^2 + |y|^2 \]

### Proof:
1. **Expand the expression for \( |x + y|^2 \):**

   The magnitude squared of a vector \( z \) is given by the dot product \( |z|^2 = \langle z, z \rangle \). So, for \( x + y \):
   \[
   |x + y|^2 = \langle x + y, x + y \rangle
   \]

2. **Apply the distributive property of the dot product:**

   \[
   \langle x + y, x + y \rangle = \langle x, x \rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle
   \]

3. **Simplify using the properties of the dot product:**

   - \( \langle x, x \rangle = |x|^2 \) (magnitude squared of \( x \))
   - \( \langle y, y \rangle = |y|^2 \) (magnitude squared of \( y \))
   - \( \langle x, y \rangle = 0 \) (given that \( x \) and \( y \) are perpendicular)
   - \( \langle y, x \rangle = \langle x, y \rangle = 0 \) (since the dot product is commutative)

   Therefore:
   \[
   \langle x + y, x + y \rangle = |x|^2 + 0 + 0 + |y|^2
   \]
   \[
   |x + y|^2 = |x|^2 + |y|^2
   \]

### Conclusion:
We have shown that if \( x \) and \( y \) are perpendicular, then \( |x + y|^2 = |x|^2 + |y|^2 \). This completes the proof.

Would you like more details, or do you have any questions?

Here are some related questions:
1. How does the dot product relate to the angle between two vectors?
2. What is the geometric interpretation of the dot product?
3. Can you prove that the magnitude of the difference of two orthogonal vectors satisfies a similar property?
4. How does the Pythagorean theorem relate to this proof in the context of vectors?
5. What happens to the expression \( |x + y|^2 \) if \( x \) and \( y \) are not perpendicular?

**Tip:** Understanding the properties of the dot product is crucial in vector algebra, especially in proofs involving orthogonal vectors.

Explore the proof that if vectors x and y in R^n are perpendicular, then |x + y|^2 = |x|^2 + |y|^2. This detailed mathematical explanation covers vector algebra concepts, the dot product, and the property of orthogonal vectors. Suitable for advanced high school and college students.

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