If `p hat(i)+3hat(j)` is a vector of magnitude 5, then the value of p is :

Options

A. 0
B. 1
C. `pm 3`
D. `pm 4`

Answer

To solve the problem, we need to find the value of \( p \) given that the vector \( \hat{p} \hat{i} + 3 \hat{j} \) has a magnitude of 5.

### Step-by-Step Solution:

1. **Define the vector**:
Let \( \vec{A} = p \hat{i} + 3 \hat{j} \).

2. **Write the formula for the magnitude of a vector**:
The magnitude of a vector \( \vec{A} = a \hat{i} + b \hat{j} \) is given by:
\[ |\vec{A}| = \sqrt{a^2 + b^2} \]
In our case, \( a = p \) and \( b = 3 \).

3. **Set up the equation for the magnitude**:
According to the problem, the magnitude of \( \vec{A} \) is 5. Therefore, we can write:
\[ |\vec{A}| = \sqrt{p^2 + 3^2} = 5 \]

4. **Square both sides to eliminate the square root**:
Squaring both sides gives:
\[ p^2 + 3^2 = 5^2 \]
This simplifies to:
\[ p^2 + 9 = 25 \]

5. **Solve for \( p^2 \)**:
Rearranging the equation, we get:
\[ p^2 = 25 - 9 \]
\[ p^2 = 16 \]

6. **Find the value of \( p \)**:
Taking the square root of both sides, we find:
\[ p = \pm 4 \]

### Final Answer:
The value of \( p \) is \( 4 \) or \( -4 \).