Which one of the following vectors is normal to the vector `hat(i)+hat(j)+hat(k)`?

Options

A. `hat(i)+hat(j)-hat(k)`
B. `hat(i)-hat(j)+hat(k)`
C. `hat(i)-hat(j)-hat(k)`
D. None of the above

Explanation

To determine which vector is normal to the vector \(\hat{i} + \hat{j} + \hat{k}\), we need to use the property of the dot product. Two vectors are normal (or perpendicular) to each other if their dot product equals zero.

Let's denote the vector \(\vec{A} = \hat{i} + \hat{j} + \hat{k}\).

Now, we will check each of the given vectors to see if their dot product with \(\vec{A}\) equals zero.

### Step 1: Check the first vector \(\hat{i} + \hat{j} - \hat{k}\)

Calculate the dot product:
\[ \vec{A} \cdot (\hat{i} + \hat{j} - \hat{k}) = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} + \hat{j} - \hat{k}) \]
\[ = 1 \cdot 1 + 1 \cdot 1 + 1 \cdot (-1) = 1 + 1 - 1 = 1 \neq 0 \]
Thus, this vector is not normal to \(\vec{A}\).

### Step 2: Check the second vector \(\hat{i} - \hat{j} + \hat{k}\)

Calculate the dot product:
\[ \vec{A} \cdot (\hat{i} - \hat{j} + \hat{k}) = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j} + \hat{k}) \]
\[ = 1 \cdot 1 + 1 \cdot (-1) + 1 \cdot 1 = 1 - 1 + 1 = 1 \neq 0 \]
Thus, this vector is also not normal to \(\vec{A}\).

### Step 3: Check the third vector \(\hat{i} - \hat{j} - \hat{k}\)

Calculate the dot product:
\[ \vec{A} \cdot (\hat{i} - \hat{j} - \hat{k}) = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j} - \hat{k}) \]
\[ = 1 \cdot 1 + 1 \cdot (-1) + 1 \cdot (-1) = 1 - 1 - 1 = -1 \neq 0 \]
Thus, this vector is also not normal to \(\vec{A}\).

### Conclusion

Since none of the given vectors are normal to \(\vec{A} = \hat{i} + \hat{j} + \hat{k}\), the answer is "none of the above."