Which of the following is perpendicular to `hati-hatj-hatk`?

Options

A. `hati+hatj+hatk`
B. `-hati+hatj+hatk`
C. `hati+hatj-hatk`
D. none of these.

Answer

To determine which vector is perpendicular to the vector \( \hat{i} - \hat{j} - \hat{k} \), we will follow these steps:

### Step 1: Define the given vector
Let the given vector be:
\[ \mathbf{A} = \hat{i} - \hat{j} - \hat{k} \]

### Step 2: Understand the condition for perpendicularity
Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular if their dot product is zero:
\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

### Step 3: Check each option
We will check the dot product of \( \mathbf{A} \) with each option provided.

#### Option A: \( \mathbf{B_1} = \hat{i} + \hat{k} + \hat{k} \)
Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B_1} = (\hat{i} - \hat{j} - \hat{k}) \cdot (\hat{i} + \hat{k} + \hat{k}) \]
\[ = \hat{i} \cdot \hat{i} + \hat{i} \cdot \hat{k} + \hat{i} \cdot \hat{k} - \hat{j} \cdot \hat{i} - \hat{j} \cdot \hat{k} - \hat{j} \cdot \hat{k} - \hat{k} \cdot \hat{i} - \hat{k} \cdot \hat{k} - \hat{k} \cdot \hat{k} \]
\[ = 1 + 0 + 0 - 0 - 0 - 0 - 0 - 1 - 1 \]
\[ = 1 - 1 - 1 = -1 \quad (\text{not perpendicular}) \]

#### Option B: \( \mathbf{B_2} = -\hat{i} + \hat{j} + \hat{k} \)
Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B_2} = (\hat{i} - \hat{j} - \hat{k}) \cdot (-\hat{i} + \hat{j} + \hat{k}) \]
\[ = -\hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j} - \hat{k} \cdot \hat{k} \]
\[ = -1 + 1 - 1 = -1 \quad (\text{not perpendicular}) \]

#### Option C: \( \mathbf{B_3} = \hat{i} + \hat{j} - \hat{k} \)
Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B_3} = (\hat{i} - \hat{j} - \hat{k}) \cdot (\hat{i} + \hat{j} - \hat{k}) \]
\[ = \hat{i} \cdot \hat{i} + \hat{j} \cdot (-\hat{j}) - \hat{k} \cdot (-\hat{k}) \]
\[ = 1 - 1 + 1 = 1 \quad (\text{not perpendicular}) \]

#### Option D: Check the remaining option
Since we have checked options A, B, and C, and they are not perpendicular, we conclude that the correct option must be D.

### Conclusion
The vector that is perpendicular to \( \hat{i} - \hat{j} - \hat{k} \) is:
\[ \text{Option D} \]