The scalar projection of vector $\mathbf{v}$ onto vector $\mathbf{u}$, denoted as $\text{comp}_{\mathbf{u}} \mathbf{v}$, is given by the formula:

$\text{comp}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|}$

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 magnitude $\|\mathbf{u}\|$ of vector $\mathbf{u}$

$\|\mathbf{u}\| = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73}$

Step 3: Compute the scalar projection

$\text{comp}_{\mathbf{u}} \mathbf{v} = \frac{39}{\sqrt{73}}$

Thus, the scalar projection $\text{comp}_{\mathbf{u}} \mathbf{v}$ is:

$\text{comp}_{\mathbf{u}} \mathbf{v} = \frac{39}{\sqrt{73}}$

This represents the length of the shadow or projection of $\mathbf{v}$ onto the direction of $\mathbf{u}$.

Would you like further explanation or have any other questions?

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Here are 5 related questions:

1. How does the scalar projection differ from the vector projection?
2. What does a negative scalar projection indicate?
3. How would you calculate the scalar projection of $\mathbf{u}$ onto $\mathbf{v}$?
4. What is the relationship between scalar projection and the cosine of the angle between two vectors?
5. How can scalar projections be used in real-world applications, such as physics or engineering?

Tip: Scalar projections help determine how much of one vector acts in the direction of another, which is useful in analyzing forces and motions in physics.