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: \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.

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!