The line whose vector equation are `vecr =2 hati - 3 hatj + 7 hatk + lamda (2 hati + p hatj + 5 hatk) and vecr = hati - 2 hatj + 3 hatk+ mu (3 hati - p hatj + p hatk)` are perpendicular for all values of `gamma and mu` if p equals :

Options

A. `-1`
B. 2
C. 5
D. 6

Answer

To find the value of \( p \) for which the two lines represented by the given vector equations are perpendicular for all values of \( \lambda \) and \( \mu \), we need to follow these steps:

### Step 1: Identify the Direction Vectors
The vector equations of the lines are given as:
1. \( \vec{r_1} = 2\hat{i} - 3\hat{j} + 7\hat{k} + \lambda(2\hat{i} + p\hat{j} + 5\hat{k}) \)
2. \( \vec{r_2} = \hat{i} - 2\hat{j} + 3\hat{k} + \mu(3\hat{i} - p\hat{j} + p\hat{k}) \)

From these equations, we can identify the direction vectors:
- For the first line, the direction vector \( \vec{b_1} = 2\hat{i} + p\hat{j} + 5\hat{k} \).
- For the second line, the direction vector \( \vec{b_2} = 3\hat{i} - p\hat{j} + p\hat{k} \).

### Step 2: Set Up the Condition for Perpendicularity
Two vectors are perpendicular if their dot product is zero. Therefore, we need to compute the dot product \( \vec{b_1} \cdot \vec{b_2} \) and set it equal to zero:
\[ \vec{b_1} \cdot \vec{b_2} = (2\hat{i} + p\hat{j} + 5\hat{k}) \cdot (3\hat{i} - p\hat{j} + p\hat{k}) = 0 \]

### Step 3: Calculate the Dot Product
Calculating the dot product:
\[ \vec{b_1} \cdot \vec{b_2} = 2 \cdot 3 + p \cdot (-p) + 5 \cdot p \]
This simplifies to:
\[ 6 - p^2 + 5p = 0 \]

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ -p^2 + 5p + 6 = 0 \]
Multiplying through by -1 to make it standard form:
\[ p^2 - 5p - 6 = 0 \]

### Step 5: Factor the Quadratic Equation
Now, we can factor the quadratic equation:
\[ (p - 6)(p + 1) = 0 \]
This gives us the solutions:
\[ p - 6 = 0 \quad \text{or} \quad p + 1 = 0 \]
Thus, we find:
\[ p = 6 \quad \text{or} \quad p = -1 \]

### Conclusion
The values of \( p \) for which the two lines are perpendicular for all values of \( \lambda \) and \( \mu \) are \( p = 6 \) and \( p = -1 \).