Given the vectors u = <1, 2, 3> and v = <2, 1, 1>. Express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v.

To express the vector \( \mathbf{u} \) as the sum \( \mathbf{u} = \mathbf{p} + \mathbf{n} \), where \( \mathbf{p} \) is parallel to \( \mathbf{v} \) and \( \mathbf{n} \) is orthogonal to \( \mathbf{v} \), we can follow these steps:

1. **Calculate the projection of \( \mathbf{u} \) onto \( \mathbf{v} \)** to find \( \mathbf{p} \):
   \[
   \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}
   \]

   - Calculate the dot products:
     \[
     \mathbf{u} \cdot \mathbf{v} = 1 \cdot 2 + 2 \cdot 1 + 3 \cdot 1 = 2 + 2 + 3 = 7
     \]
     \[
     \mathbf{v} \cdot \mathbf{v} = 2 \cdot 2 + 1 \cdot 1 + 1 \cdot 1 = 4 + 1 + 1 = 6
     \]

   - Now calculate the projection:
     \[
     \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{7}{6} \mathbf{v} = \frac{7}{6} \langle 2, 1, 1 \rangle = \left\langle \frac{14}{6}, \frac{7}{6}, \frac{7}{6} \right\rangle = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle
     \]

   So, \( \mathbf{p} = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle \).

2. **Calculate \( \mathbf{n} \)**:
   \[
   \mathbf{n} = \mathbf{u} - \mathbf{p} = \langle 1, 2, 3 \rangle - \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle
   \]
   - Perform the subtraction:
     \[
     \mathbf{n} = \left\langle 1 - \frac{7}{3}, 2 - \frac{7}{6}, 3 - \frac{7}{6} \right\rangle
     \]
     \[
     = \left\langle \frac{3}{3} - \frac{7}{3}, \frac{12}{6} - \frac{7}{6}, \frac{18}{6} - \frac{7}{6} \right\rangle = \left\langle -\frac{4}{3}, \frac{5}{6}, \frac{11}{6} \right\rangle
     \]

Thus, the decomposition of \( \mathbf{u} \) is:
\[
\mathbf{u} = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle + \left\langle -\frac{4}{3}, \frac{5}{6}, \frac{11}{6} \right\rangle
\]

### Summary:
- \( \mathbf{p} = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle \) (parallel to \( \mathbf{v} \))
- \( \mathbf{n} = \left\langle -\frac{4}{3}, \frac{5}{6}, \frac{11}{6} \right\rangle \) (orthogonal to \( \mathbf{v} \))

Would you like further details or have any questions? Here are some follow-up questions to consider:

1. How do you calculate the angle between two vectors?
2. What are the geometric interpretations of parallel and orthogonal vectors?
3. Can you derive the formula for the projection of one vector onto another?
4. What properties do orthogonal vectors have in a vector space?
5. How would you generalize this method for higher dimensions? 

**Tip:** Always check that the dot product of \( \mathbf{n} \) and \( \mathbf{v} \) is zero to confirm orthogonality.

Explore how to express the vector u = <1, 2, 3> as the sum of a component p parallel to v = <2, 1, 1> and a component n orthogonal to v. This detailed solution covers vector projection and orthogonality concepts, making it suitable for high school students in Grades 10-12.

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