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The question in the image asks: *"Which of the following represents the angle between the vectors \( \vec{i} + \vec{j} + \vec{k} \) and \( \vec{i} + \vec{j} - \vec{k} \)?"*

The approach provided uses the formula for the angle between two vectors \( \vec{a} \) and \( \vec{b} \) in three dimensions:

\[
\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}
\]

### Step-by-step process:

1. **Dot product \( \vec{a} \cdot \vec{b} \):**
   \[
   \vec{a} = \vec{i} + \vec{j} + \vec{k}, \quad \vec{b} = \vec{i} + \vec{j} - \vec{k}
   \]
   The dot product of these two vectors is:
   \[
   \vec{a} \cdot \vec{b} = (1 \cdot 1) + (1 \cdot 1) + (1 \cdot (-1)) = 1 + 1 - 1 = 1
   \]

2. **Magnitude of the vectors \( |\vec{a}| \) and \( |\vec{b}| \):**
   The magnitude of both vectors is the same:
   \[
   |\vec{a}| = |\vec{b}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}
   \]

3. **Cosine of the angle \( \theta \):**
   Using the dot product and the magnitudes:
   \[
   \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \frac{1}{\sqrt{3} \cdot \sqrt{3}} = \frac{1}{3}
   \]

4. **Angle \( \theta \):**
   Therefore, the angle is:
   \[
   \theta = \cos^{-1}\left(\frac{1}{3}\right)
   \]

### Correct Answer:
The answer is **Option B: \( \cos^{-1}\left( \frac{1}{3} \right) \)**.

Would you like further explanation or more details?

### Follow-up Questions:
1. How do you compute the dot product of two vectors in 3D?
2. How is the magnitude of a vector determined?
3. What does the angle between two vectors represent geometrically?
4. Why is the cosine formula used to find the angle between vectors?
5. How can the result \( \cos^{-1}(1/3) \) be converted into degrees?

### Tip:
When computing the angle between vectors, always ensure that the magnitudes of the vectors are correctly calculated before using the cosine formula.

Which of the following represents the angle between the vectors i + j + k and i + j - k?

Learn how to find the angle between two vectors i + j + k and i + j - k using the dot product and cosine formula. This solution involves calculating the dot product, magnitudes of vectors, and solving for the angle θ. Suitable for high school and early college students.

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