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**v)** Let \( u = i + n(j - i) \) and \( v = i - j \), where \( n \) is a constant. Find the angle(s) between \( u \) and \( v \).

This question is asking you to find the angle between two vectors \( u \) and \( v \), given their definitions in terms of other vectors \( i \) and \( j \).

### Solution:

1. **Vectors Given:**
   \[
   u = i + n(j - i) = (1 - n)i + nj
   \]
   \[
   v = i - j
   \]

2. **Dot Product Formula:**
   The angle \( \theta \) between two vectors \( u \) and \( v \) can be found using the dot product:
   \[
   \cos(\theta) = \frac{u \cdot v}{\|u\| \|v\|}
   \]
   Where:
   - \( u \cdot v \) is the dot product of the two vectors
   - \( \|u\| \) and \( \|v\| \) are the magnitudes (norms) of the vectors.

3. **Dot Product:**
   \[
   u \cdot v = ((1 - n)i + nj) \cdot (i - j)
   \]
   Expanding the dot product:
   \[
   u \cdot v = (1 - n)(i \cdot i) + (1 - n)(-i \cdot j) + n(j \cdot i) + (-n)(j \cdot j)
   \]
   Simplifying:
   \[
   u \cdot v = (1 - n)(1) + (-n)(1)
   \]
   Since \( i \cdot j = 0 \) and \( i \cdot i = j \cdot j = 1 \), we get:
   \[
   u \cdot v = 1 - n - n = 1 - 2n
   \]

4. **Magnitudes:**
   Now, calculate the magnitudes \( \|u\| \) and \( \|v\| \).

   - Magnitude of \( u \):
     \[
     \|u\| = \sqrt{((1 - n)i + nj) \cdot ((1 - n)i + nj)} = \sqrt{(1 - n)^2 + n^2}
     \]
     Simplifying:
     \[
     \|u\| = \sqrt{(1 - 2n + n^2 + n^2)} = \sqrt{1 - 2n + 2n^2}
     \]

   - Magnitude of \( v \):
     \[
     \|v\| = \sqrt{(i - j) \cdot (i - j)} = \sqrt{1 + 1} = \sqrt{2}
     \]

5. **Angle Between \( u \) and \( v \):**
   Now, use the dot product and magnitudes to find \( \cos(\theta) \):
   \[
   \cos(\theta) = \frac{1 - 2n}{\sqrt{1 - 2n + 2n^2} \cdot \sqrt{2}}
   \]

   This gives you the cosine of the angle between \( u \) and \( v \). The angle \( \theta \) can then be calculated using:
   \[
   \theta = \cos^{-1}\left(\frac{1 - 2n}{\sqrt{2(1 - 2n + 2n^2)}}\right)
   \]

Let me know if you'd like to explore the solution in more detail or have any questions!

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Here are 5 related questions for further exploration:
1. How do you calculate the angle between vectors in 3D space?
2. What is the geometric interpretation of the dot product?
3. How do the values of \( n \) affect the angle between \( u \) and \( v \)?
4. What is the significance of perpendicular vectors in vector geometry?
5. How does the cross product differ from the dot product when dealing with vectors?

**Tip:** Understanding the dot product helps in many areas of physics and geometry, especially when analyzing angles between vectors!

Let u = i + n(j - i) and v = i - j, where n is a constant. Find the angles between u and v.

Learn how to find the angle between vectors u and v, where u = i + n(j - i) and v = i - j, using the dot product and magnitude formulas. Explore the solution with a step-by-step breakdown suitable for high school and college students.

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