If a, b,c are the pth, qth, and rth terms of a HP, then the vectors ` vecu= hati/a + hatj/b + hatk/c and vecv = ( q -r) hati + ( r-p) hatj + ( p-q) hatk`

Options

A. are parallel
B. are othogonal
C. satisfy `vecu .vecv =1`
D. satisfy `|vecu xx vecv| =hati +hatj +hatk`

Explanation

To solve the problem, we need to show that the vectors \( \vec{u} \) and \( \vec{v} \) are orthogonal, which means their dot product is zero.

### Step-by-Step Solution:

1. **Understanding the terms in Harmonic Progression (HP)**:
- The terms \( a, b, c \) are the \( p^{th}, q^{th}, r^{th} \) terms of a Harmonic Progression (HP).
- In an HP, the reciprocals of the terms form an Arithmetic Progression (AP).

2. **Expressing the terms in HP**:
- Let \( A \) be the first term of the corresponding AP and \( D \) be the common difference.
- Thus, we can express the terms as:
\[ \frac{1}{a} = A + (p-1)D \]
\[ \frac{1}{b} = A + (q-1)D \]
\[ \frac{1}{c} = A + (r-1)D \]

3. **Finding the expressions for \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \)**:
- Rearranging the above equations gives:
\[ \frac{1}{a} = A + (p-1)D \]
\[ \frac{1}{b} = A + (q-1)D \]
\[ \frac{1}{c} = A + (r-1)D \]

4. **Substituting into vector \( \vec{u} \)**:
- The vector \( \vec{u} \) can be expressed as:
\[ \vec{u} = \hat{i} \frac{1}{a} + \hat{j} \frac{1}{b} + \hat{k} \frac{1}{c} \]
- Substituting the expressions for \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \):
\[ \vec{u} = \hat{i}(A + (p-1)D) + \hat{j}(A + (q-1)D) + \hat{k}(A + (r-1)D) \]

5. **Substituting into vector \( \vec{v} \)**:
- The vector \( \vec{v} \) is given as:
\[ \vec{v} = (q - r) \hat{i} + (r - p) \hat{j} + (p - q) \hat{k} \]

6. **Calculating the dot product \( \vec{u} \cdot \vec{v} \)**:
- The dot product is calculated as:
\[ \vec{u} \cdot \vec{v} = \left( \frac{1}{a} (q - r) + \frac{1}{b} (r - p) + \frac{1}{c} (p - q) \right) \]
- Substituting the values of \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \):
\[ \vec{u} \cdot \vec{v} = \left( A + (p-1)D \right)(q - r) + \left( A + (q-1)D \right)(r - p) + \left( A + (r-1)D \right)(p - q) \]

7. **Simplifying the expression**:
- After simplification, we find that:
\[ \vec{u} \cdot \vec{v} = 0 \]
- This indicates that the vectors \( \vec{u} \) and \( \vec{v} \) are orthogonal.

### Conclusion:
Thus, we conclude that the vectors \( \vec{u} \) and \( \vec{v} \) are orthogonal.