To find the vector projection $\mathbf{w} = \text{proj}_{\mathbf{u}} \mathbf{v}$ of vector $\mathbf{v}$ onto vector $\mathbf{u}$, we can use the formula:

$\mathbf{w} = \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$

Given: $\mathbf{u} = 8\mathbf{i} + 3\mathbf{j}, \quad \mathbf{v} = 3\mathbf{i} + 5\mathbf{j}$

Step 1: Calculate the dot product $\mathbf{v} \cdot \mathbf{u}$

$\mathbf{v} \cdot \mathbf{u} = (3)(8) + (5)(3) = 24 + 15 = 39$

Step 2: Calculate the dot product $\mathbf{u} \cdot \mathbf{u}$

$\mathbf{u} \cdot \mathbf{u} = (8)(8) + (3)(3) = 64 + 9 = 73$

Step 3: Compute the projection

$\mathbf{w} = \frac{39}{73} \mathbf{u} = \frac{39}{73} (8\mathbf{i} + 3\mathbf{j})$

Step 4: Express $\mathbf{w}$ in component form

$\mathbf{w} = \left(\frac{39 \times 8}{73}\right) \mathbf{i} + \left(\frac{39 \times 3}{73}\right) \mathbf{j}$ $\mathbf{w} = \left(\frac{312}{73}\right) \mathbf{i} + \left(\frac{117}{73}\right) \mathbf{j}$

Thus, the vector projection $\mathbf{w}$ is:

$\mathbf{w} = \frac{312}{73} \mathbf{i} + \frac{117}{73} \mathbf{j}$

This is the vector projection of $\mathbf{v}$ onto $\mathbf{u}$ in component form.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you calculate the magnitude of a vector?
2. What is the geometric interpretation of a vector projection?
3. How would you find the orthogonal projection of $\mathbf{v}$ onto $\mathbf{u}$?
4. Can vector projections be used in finding the angle between two vectors?
5. How do vector projections relate to the concept of work done by a force?

Tip: Vector projections are useful in physics, especially when decomposing forces into components along specific directions.