https://file.aiquickdraw.com/mathbot/file/1728560608235hem3h3l9.jpg

Let's analyze the problem from the image. The questions we have are as follows:

1. **Part (s):** 
   The task is to find the angle between the vectors:
   \[
   4\mathbf{i} - \mathbf{j} + \sqrt{3}\mathbf{k} \quad \text{and} \quad 2\mathbf{i} - 5\mathbf{k}
   \]
   This can be done using the formula for the angle between two vectors \( \theta \):
   \[
   \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}
   \]
   where \( \mathbf{a} \) and \( \mathbf{b} \) are the vectors, \( \mathbf{a} \cdot \mathbf{b} \) is the dot product, and \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes of the vectors.

2. **Part (r):**
   Given vectors:
   \[
   \mathbf{a} = 2\mathbf{i} + 5\mathbf{j} - 7\mathbf{k}
   \]
   \[
   \mathbf{b} = 3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}
   \]
   \[
   \mathbf{c} = 4\mathbf{i} + 10\mathbf{j} - 14\mathbf{k}
   \]
   The question asks to show that \( \mathbf{a} \) is parallel to \( \mathbf{c} \) but is perpendicular to \( \mathbf{b} \).

Would you like a detailed step-by-step solution for each part?

Here are some questions to think about while we go through the solution:

1. What is the formula for the dot product of two vectors?
2. How do you determine if two vectors are parallel using their components?
3. What conditions must be met for two vectors to be perpendicular?
4. What are the magnitudes of the given vectors in part (s)?
5. How do we interpret the cosine of the angle when it equals zero or one?

**Tip:** Parallel vectors have proportional components, while perpendicular vectors have a dot product of zero.

Consider the vectors a = 2i + 5j - 7k, b = 3i + 3j + 3k, and c = 4i + 10j - 14k. Show that a is parallel to c but is perpendicular to b. Also, calculate the angle between the vectors 4i - j + √3k and 2i - 5k.

Learn how to show that two vectors are parallel or perpendicular using dot products and vector analysis. Includes angle calculation between vectors using trigonometry. This problem covers vector algebra, dot products, and key geometric properties of vectors.

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