For any two non-zero vectors a and b, |a|b+|b|a and |a|b-|b|a are

Options

A. parallel
B. perpendicular
C. non-parallel
D. None of these

Explanation

To solve the problem, we need to analyze the two vectors given: \( p = |a|b + |b|a \) and \( q = |a|b - |b|a \). We will determine whether these vectors are parallel, perpendicular, or non-parallel by calculating their dot product.

### Step 1: Define the vectors
Let:
- \( p = |a|b + |b|a \)
- \( q = |a|b - |b|a \)

### Step 2: Calculate the dot product \( p \cdot q \)
Using the definition of the dot product, we have:
\[ p \cdot q = (|a|b + |b|a) \cdot (|a|b - |b|a) \]

### Step 3: Expand the dot product
We can expand this using the distributive property of the dot product:
\[ p \cdot q = |a|b \cdot |a|b - |a|b \cdot |b|a + |b|a \cdot |a|b - |b|a \cdot |b|a \]

### Step 4: Simplify each term
1. The first term is:
\[ |a|b \cdot |a|b = |a|^2 (b \cdot b) = |a|^2 |b|^2 \]
2. The second term is:
\[ -|a|b \cdot |b|a = -|b| (b \cdot a) |a| \]
3. The third term is:
\[ |b|a \cdot |a|b = |a| (a \cdot b) |b| \]
4. The fourth term is:
\[ -|b|a \cdot |b|a = -|b|^2 (a \cdot a) = -|b|^2 |a|^2 \]

### Step 5: Combine the terms
Now, we combine all these terms:
\[ p \cdot q = |a|^2 |b|^2 - |b| (b \cdot a) |a| + |a| (a \cdot b) |b| - |b|^2 |a|^2 \]
Notice that \( - |b| (b \cdot a) |a| + |a| (a \cdot b) |b| = 0 \) because \( b \cdot a = a \cdot b \).

Thus, we have:
\[ p \cdot q = |a|^2 |b|^2 - |b|^2 |a|^2 = 0 \]

### Step 6: Conclusion
Since the dot product \( p \cdot q = 0 \), this implies that the vectors \( p \) and \( q \) are perpendicular.

### Final Answer
The vectors \( |a|b + |b|a \) and \( |a|b - |b|a \) are **perpendicular**.

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