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To find the projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\), we use the formula:

\[
\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}
\]

where:

- \(\mathbf{u} = \begin{bmatrix} 5 \\ 3 \\ 0 \end{bmatrix}\)
- \(\mathbf{v} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}\)

First, we need to calculate the dot product \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{v}\).

### Step 1: Compute the dot products

1. **Dot product \(\mathbf{u} \cdot \mathbf{v}\):**

Since \(\mathbf{u}\) and \(\mathbf{v}\) have different dimensions (3 and 2 components respectively), it is not possible to compute the projection directly. Instead, we need to ensure that both vectors have the same dimension or clarify the dimensions needed.

Can you please confirm the correct dimensions for \(\mathbf{u}\) and \(\mathbf{v}\)?

Learn how to calculate the vector projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) using the dot product formula. This problem involves vectors with specific dimensions and requires careful computation. Suitable for advanced high school students.

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