The magnitude of the scalar p for which the vector `p(-3hat(i) -2hat(j) +13hat(k))` is of unit length is

Options

A. a)1//8
B. b) 1//64
C. c) sqrt182
D. d) 1//sqrt182

Answer

To find the magnitude of the scalar \( p \) for which the vector \( p(-3\hat{i} - 2\hat{j} + 13\hat{k}) \) is of unit length, we can follow these steps:

### Step 1: Define the vector
The given vector can be expressed as:
\[ \mathbf{v} = p(-3\hat{i} - 2\hat{j} + 13\hat{k}) = -3p\hat{i} - 2p\hat{j} + 13p\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
For our vector:
\[ |\mathbf{v}| = \sqrt{(-3p)^2 + (-2p)^2 + (13p)^2} \]

### Step 3: Substitute and simplify
Substituting the components:
\[ |\mathbf{v}| = \sqrt{(9p^2) + (4p^2) + (169p^2)} \]
Combine the terms:
\[ |\mathbf{v}| = \sqrt{(9 + 4 + 169)p^2} = \sqrt{182p^2} \]

### Step 4: Set the magnitude equal to 1
Since we want the vector to be of unit length, we set the magnitude equal to 1:
\[ \sqrt{182p^2} = 1 \]

### Step 5: Square both sides
Squaring both sides to eliminate the square root gives:
\[ 182p^2 = 1 \]

### Step 6: Solve for \( p^2 \)
Now, we can solve for \( p^2 \):
\[ p^2 = \frac{1}{182} \]

### Step 7: Solve for \( p \)
Taking the square root of both sides, we find:
\[ p = \sqrt{\frac{1}{182}} = \frac{1}{\sqrt{182}} \]

### Final Answer
Thus, the magnitude of the scalar \( p \) is:
\[ p = \frac{1}{\sqrt{182}} \]