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To find the vector projection of **u** onto **v**, we use the formula:

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

Given:
- \(\mathbf{u} = \langle 3, 5, -1 \rangle\)
- \(\mathbf{v} = \langle 8, -6, 10 \rangle\)

### Step-by-step Solution:

1. Compute the dot product \(\mathbf{u} \cdot \mathbf{v}\):

\[
\mathbf{u} \cdot \mathbf{v} = 3 \times 8 + 5 \times (-6) + (-1) \times 10 = 24 - 30 - 10 = -16
\]

2. Compute the dot product \(\mathbf{v} \cdot \mathbf{v}\) (magnitude squared of **v**):

\[
\mathbf{v} \cdot \mathbf{v} = 8^2 + (-6)^2 + 10^2 = 64 + 36 + 100 = 200
\]

3. Compute the scalar factor for the projection:

\[
\frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} = \frac{-16}{200} = -0.08
\]

4. Multiply this scalar by vector **v** to find the projection:

\[
\text{proj}_{\mathbf{v}} \mathbf{u} = -0.08 \times \langle 8, -6, 10 \rangle = \langle -0.64, 0.48, -0.8 \rangle
\]

### Result:
\[
\text{proj}_{\mathbf{v}} \mathbf{u} = \langle -0.64, 0.48, -0.8 \rangle
\]

Would you like more detailed explanations or have any questions?

### Related Questions:
1. How do you find the dot product of two vectors in 3D space?
2. What is the geometric interpretation of the vector projection?
3. How do you find the magnitude of a vector in three-dimensional space?
4. How would the projection change if vector \(\mathbf{u}\) was parallel to vector \(\mathbf{v}\)?
5. Can you find the orthogonal component of vector \(\mathbf{u}\) relative to \(\mathbf{v}\)?

### Tip:
When finding vector projections, if \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\), the projection will be the zero vector.

Given u = <3, 5, -1> and v = <8, -6, 10>, find the vector proj_v(u).

Learn how to find the vector projection of u = <3, 5, -1> onto v = <8, -6, 10> using the projection formula. This solution includes calculations of the dot product and scalar factor for the projection, providing a step-by-step approach for high school students studying vector operations.

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