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The problem in the image asks to show that the vector \( \mathbf{a} = 2\mathbf{i} + 5\mathbf{j} - 7\mathbf{k} \) is parallel to vector \( \mathbf{c} = 4\mathbf{i} + 10\mathbf{j} - 14\mathbf{k} \) but is perpendicular to vector \( \mathbf{b} = 3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k} \).

### Solution:

#### Step 1: Check if \( \mathbf{a} \) is parallel to \( \mathbf{c} \)
Two vectors are parallel if one is a scalar multiple of the other. In other words, there must exist a constant \( \lambda \) such that:
\[
\mathbf{a} = \lambda \mathbf{c}
\]

Comparing the components of \( \mathbf{a} = 2\mathbf{i} + 5\mathbf{j} - 7\mathbf{k} \) and \( \mathbf{c} = 4\mathbf{i} + 10\mathbf{j} - 14\mathbf{k} \):

- \( 2 = \lambda \cdot 4 \Rightarrow \lambda = \frac{2}{4} = \frac{1}{2} \)
- \( 5 = \lambda \cdot 10 \Rightarrow \lambda = \frac{5}{10} = \frac{1}{2} \)
- \( -7 = \lambda \cdot (-14) \Rightarrow \lambda = \frac{-7}{-14} = \frac{1}{2} \)

Since the same scalar \( \lambda = \frac{1}{2} \) works for all components, \( \mathbf{a} \) is parallel to \( \mathbf{c} \).

#### Step 2: Check if \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \)
Two vectors are perpendicular if their dot product is zero. The dot product of \( \mathbf{a} = 2\mathbf{i} + 5\mathbf{j} - 7\mathbf{k} \) and \( \mathbf{b} = 3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k} \) is calculated as:
\[
\mathbf{a} \cdot \mathbf{b} = (2)(3) + (5)(3) + (-7)(3)
\]
\[
= 6 + 15 - 21 = 0
\]

Since the dot product is zero, \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \).

### Conclusion:
- \( \mathbf{a} \) is parallel to \( \mathbf{c} \).
- \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \).

Would you like more details or have any questions?

### Related Questions:
1. What are the conditions for two vectors to be parallel in 3D space?
2. How do you compute the dot product of two vectors?
3. What is the geometric interpretation of the cross product between two vectors?
4. How can you find the angle between two vectors using their dot product?
5. What does it mean if the dot product between two vectors is positive, negative, or zero?

### Tip:
When dealing with vectors, always check if two vectors are parallel by comparing their components and checking for a constant ratio.

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.

Learn how to show that vector a = 2i + 5j - 7k is parallel to vector c = 4i + 10j - 14k, but is perpendicular to vector b = 3i + 3j + 3k. The solution includes concepts like parallel vectors, dot product calculations, and vector algebra. Suitable for students in Grades 11-12.

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