Consider vectors u and v.
u = 8i + 3j,  v = 3i + 5j
(a)
Find the vector projection
w = projuv
of vector v onto vector u. Express your answer in component form.
w =

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.

Learn how to find the vector projection of vector v onto vector u using the dot product and vector projection formula. Detailed step-by-step solution with component form calculations included.

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