Given `vec(P)=3hat(i)-4hat(j)`. Which of the following is perpendicular to `vec(P)`?

Options

A. `3hat(i)`
B. `4hat(j)`
C. `4hat(i)+3hat(j)`
D. `4hat(i)-3hat(j)`

Explanation

To determine which vector is perpendicular to the given vector \(\vec{P} = 3\hat{i} - 4\hat{j}\), we will use the property that two vectors are perpendicular if their dot product is equal to zero.

### Step-by-Step Solution:

1. **Understand the Given Vector**:
The vector \(\vec{P}\) is given as:
\[ \vec{P} = 3\hat{i} - 4\hat{j} \]

2. **Identify the Condition for Perpendicularity**:
For two vectors \(\vec{A}\) and \(\vec{B}\) to be perpendicular, the dot product must satisfy:
\[ \vec{A} \cdot \vec{B} = 0 \]

3. **Calculate the Dot Product with Each Option**:
We will check the dot product of \(\vec{P}\) with each of the given options.

- **Option 1**: Let’s denote this vector as \(\vec{Q_1} = 3\hat{i}\).
\[ \vec{P} \cdot \vec{Q_1} = (3\hat{i} - 4\hat{j}) \cdot (3\hat{i}) = 3 \cdot 3 + (-4) \cdot 0 = 9 \quad (\text{not } 0) \]

- **Option 2**: Let’s denote this vector as \(\vec{Q_2} = 4\hat{j}\).
\[ \vec{P} \cdot \vec{Q_2} = (3\hat{i} - 4\hat{j}) \cdot (4\hat{j}) = 3 \cdot 0 + (-4) \cdot 4 = -16 \quad (\text{not } 0) \]

- **Option 3**: Let’s denote this vector as \(\vec{Q_3} = 4\hat{i} + 3\hat{j}\).
\[ \vec{P} \cdot \vec{Q_3} = (3\hat{i} - 4\hat{j}) \cdot (4\hat{i} + 3\hat{j}) = 3 \cdot 4 + (-4) \cdot 3 = 12 - 12 = 0 \quad (\text{is } 0) \]

4. **Conclusion**:
The vector that is perpendicular to \(\vec{P}\) is:
\[ \vec{Q_3} = 4\hat{i} + 3\hat{j} \]

### Final Answer:
The vector that is perpendicular to \(\vec{P}\) is \(4\hat{i} + 3\hat{j}\).

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