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In this solution, the projection of a vector **v** onto the subspace \( W \), spanned by two orthogonal vectors \( \mathbf{w_1} \) and \( \mathbf{w_2} \), is **the sum of its projections onto the two orthogonal components**. This process leverages the **property of orthogonal projections**.

Here’s a breakdown of why you add the projections:

1. **Orthogonal vectors don't interfere**:
   - Since \( \mathbf{w_1} \cdot \mathbf{w_2} = 0 \), the two vectors \( \mathbf{w_1} \) and \( \mathbf{w_2} \) span independent directions in the subspace \( W \). The projection of **v** onto \( W \) needs to account for the contribution from both orthogonal components.

2. **Projection decomposes vector components**:
   - To get the **full projection of \( \mathbf{v} \) onto the subspace \( W \)**, you first project \( \mathbf{v} \) onto each basis vector:
     \[
     \text{Proj}_{\mathbf{w_1}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w_1}}{\mathbf{w_1} \cdot \mathbf{w_1}} \mathbf{w_1}
     \quad \text{and} \quad
     \text{Proj}_{\mathbf{w_2}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w_2}}{\mathbf{w_2} \cdot \mathbf{w_2}} \mathbf{w_2}.
     \]

3. **Adding results gives the total projection**:
   - Since the two vectors are orthogonal, the total projection of \( \mathbf{v} \) onto \( W \) is simply:
     \[
     \text{Proj}_{W} \mathbf{v} = \text{Proj}_{\mathbf{w_1}} \mathbf{v} + \text{Proj}_{\mathbf{w_2}} \mathbf{v}.
     \]
   - Each projection captures how much of **v** lies in the direction of \( \mathbf{w_1} \) and \( \mathbf{w_2} \), respectively. The sum then reflects the complete projection of **v** onto the plane or subspace spanned by these vectors.

This idea works only because \( \mathbf{w_1} \) and \( \mathbf{w_2} \) are orthogonal. If they weren’t orthogonal, the projections would not be independent, and a more complicated process involving Gram-Schmidt orthogonalization would be required.

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### Do you want further details or have any questions?

Here are five related questions to deepen your understanding:
1. How does the orthogonality of two vectors affect their projections?
2. What would happen if \( \mathbf{w_1} \) and \( \mathbf{w_2} \) weren’t orthogonal?
3. Can you explain the geometric intuition behind the sum of projections?
4. How does projection change when dealing with higher-dimensional spaces?
5. How does the result differ if \( \mathbf{v} \) is not in the span of \( \mathbf{w_1} \) and \( \mathbf{w_2} \)?

**Tip:** Always check for orthogonality when working with projections, as it makes the computations much simpler!

Understand why projections of a vector onto two orthogonal vectors are added. This problem covers key linear algebra concepts, including vector projections, orthogonal vectors, and the orthogonal projection theorem. Ideal for students in undergraduate linear algebra courses.

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