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Let's work through the problem step by step. We have three vectors:

\[
u = \begin{bmatrix} 5 \\ 4 \\ 1 \end{bmatrix}, \quad v = \begin{bmatrix} 3 \\ 4 \\ 1 \end{bmatrix}, \quad w = \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}
\]

### Part (a): Compute the generalized cosine between all possible pairs.
The cosine of the angle between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by the formula:

\[
\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}
\]

#### Pairs to consider:
- \( u \) and \( v \)
- \( u \) and \( w \)
- \( v \) and \( w \)

Let's compute the dot products and magnitudes for each pair.

### Part (b): Which pair of vectors is most collinear?
Collinear vectors have a cosine close to 1 or -1 (indicating they are in the same or opposite directions).

### Part (c): Which pair are most orthogonal?
Orthogonal vectors have a cosine of 0 (i.e., their dot product is zero).

### Part (d): Are \( u \), \( v \), and \( w \) linearly independent?
To check if the vectors are linearly independent, we can determine if the determinant of the matrix formed by \( u \), \( v \), and \( w \) as columns is non-zero. If the determinant is zero, they are linearly dependent.

Let me calculate these values.Here are the results for each part of the problem:

### (a) Generalized Cosine between all pairs:
- \( \cos(u, v) = 0.968 \)
- \( \cos(u, w) = 0.0 \)
- \( \cos(v, w) = -0.105 \)

### (b) Most collinear pair:
The pair \( u \) and \( v \) are the most collinear, as their cosine value is closest to 1 (\( \cos(u, v) = 0.968 \)).

### (c) Most orthogonal pair:
The pair \( u \) and \( w \) are most orthogonal, as their cosine value is \( 0 \), indicating they are perpendicular.

### (d) Linear independence:
The determinant of the matrix formed by \( u \), \( v \), and \( w \) is \( 28 \), which is non-zero. This means the vectors \( u \), \( v \), and \( w \) are linearly independent.

Would you like further details on any part? Here are five follow-up questions you may find interesting:

1. How would the results change if one of the vectors were scaled by a constant?
2. Can we visualize the relationship between the vectors in 3D space?
3. What does it mean geometrically for vectors to be collinear or orthogonal?
4. How would you verify linear independence using other methods, like row reduction?
5. How does the determinant help determine linear dependence or independence?

**Tip:** When calculating the cosine between vectors, ensure their magnitudes are non-zero to avoid division by zero errors.

Let u = [5, 4, 1]^T, v = [3, 4, 1]^T, w = [1, -2, 3]^T. (a) Compute the generalized cosine between all possible distinct pairs. (b) Which pair of vectors is most collinear? (c) Which pair is most orthogonal? (d) Are u, v, and w linearly independent?

Learn how to calculate the cosine similarity between vectors u, v, and w. Explore which vectors are most collinear or orthogonal and determine their linear independence using determinants. This problem covers essential vector analysis concepts in linear algebra.

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