Find the scalar projection 

comp**uv**

 of vector **v** onto vector **u**.

comp**uv** =

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.

Learn how to find the scalar projection of vector v onto vector u with detailed steps and a formula explanation. This problem involves vectors, dot product calculation, and vector magnitude. Suitable for high school students and anyone studying vector mathematics.

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