`p_1 hati+p_2hatj` is a unit vector perpendicular to `4hati-3hatj` if

Options

A. `p_(1)=0.6,p_(2)=0.8`
B. `p_(1)=3,p_(2)=4`
C. `p_(1)=0.8,p_(2)=0.6`
D. `p_(1)=4, p_(2)=3`

Answer

To solve the problem, we need to determine the values of \( p_1 \) and \( p_2 \) such that the vector \( p_1 \hat{i} + p_2 \hat{j} \) is a unit vector perpendicular to the vector \( 4 \hat{i} - 3 \hat{j} \).

### Step 1: Set up the dot product condition for perpendicular vectors
Since two vectors are perpendicular if their dot product is zero, we can set up the equation:
\[ (p_1 \hat{i} + p_2 \hat{j}) \cdot (4 \hat{i} - 3 \hat{j}) = 0 \]

### Step 2: Calculate the dot product
Calculating the dot product gives:
\[ p_1 \cdot 4 + p_2 \cdot (-3) = 0 \]
This simplifies to:
\[ 4p_1 - 3p_2 = 0 \]

### Step 3: Rearranging the equation
From the equation \( 4p_1 - 3p_2 = 0 \), we can express \( p_2 \) in terms of \( p_1 \):
\[ 4p_1 = 3p_2 \implies p_2 = \frac{4}{3}p_1 \]

### Step 4: Set up the unit vector condition
Since \( p_1 \hat{i} + p_2 \hat{j} \) is a unit vector, we need to satisfy the condition:
\[ \sqrt{p_1^2 + p_2^2} = 1 \]
Squaring both sides gives:
\[ p_1^2 + p_2^2 = 1 \]

### Step 5: Substitute \( p_2 \) in the unit vector equation
Substituting \( p_2 = \frac{4}{3}p_1 \) into the unit vector equation:
\[ p_1^2 + \left(\frac{4}{3}p_1\right)^2 = 1 \]
This expands to:
\[ p_1^2 + \frac{16}{9}p_1^2 = 1 \]
Combining the terms gives:
\[ \left(1 + \frac{16}{9}\right)p_1^2 = 1 \]
\[ \frac{25}{9}p_1^2 = 1 \]

### Step 6: Solve for \( p_1^2 \)
Multiplying both sides by \( \frac{9}{25} \):
\[ p_1^2 = \frac{9}{25} \]
Taking the square root gives:
\[ p_1 = \pm \frac{3}{5} = \pm 0.6 \]

### Step 7: Find \( p_2 \)
Using \( p_2 = \frac{4}{3}p_1 \):
- If \( p_1 = 0.6 \):
\[ p_2 = \frac{4}{3} \times 0.6 = 0.8 \]
- If \( p_1 = -0.6 \):
\[ p_2 = \frac{4}{3} \times (-0.6) = -0.8 \]

### Step 8: Conclusion
Since we are only interested in positive values of \( p_1 \) and \( p_2 \), we conclude:
\[ p_1 = 0.6, \quad p_2 = 0.8 \]

Thus, the correct option is:
**Option 1: \( p_1 = 0.6 \) and \( p_2 = 0.8 \)**.

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