If `vec(P)=hat(i)+hat(j)-hat(k)` and `vec(Q)=hat(i)-hat(j)+hat(k)`, then unit vector along `(vec(P)-vec(Q))` is `:`

Options

A. `(1)/(sqrt(2))hat(i)-(1)/(2)hat(k)`
B. `(sqrt(2)hat(j)-sqrt(2)hat(k))/(2)`
C. `(hat(j)-hat(k))/(2sqrt(2))`
D. `(2hat(j)-2hat(k))/(4)`

Explanation

To find the unit vector along the vector \( \vec{P} - \vec{Q} \), we can follow these steps:

### Step 1: Write down the vectors
We are given:
\[ \vec{P} = \hat{i} + \hat{j} - \hat{k} \]
\[ \vec{Q} = \hat{i} - \hat{j} + \hat{k} \]

### Step 2: Calculate \( \vec{P} - \vec{Q} \)
Now, we will subtract \( \vec{Q} \) from \( \vec{P} \):
\[ \vec{P} - \vec{Q} = (\hat{i} + \hat{j} - \hat{k}) - (\hat{i} - \hat{j} + \hat{k}) \]
Distributing the negative sign:
\[ = \hat{i} + \hat{j} - \hat{k} - \hat{i} + \hat{j} - \hat{k} \]
Now, combine like terms:
- The \( \hat{i} \) terms: \( \hat{i} - \hat{i} = 0 \)
- The \( \hat{j} \) terms: \( \hat{j} + \hat{j} = 2\hat{j} \)
- The \( \hat{k} \) terms: \( -\hat{k} - \hat{k} = -2\hat{k} \)

Thus, we have:
\[ \vec{P} - \vec{Q} = 2\hat{j} - 2\hat{k} \]

### Step 3: Simplify the result
We can factor out a 2:
\[ \vec{P} - \vec{Q} = 2(\hat{j} - \hat{k}) \]

### Step 4: Find the magnitude of \( \vec{P} - \vec{Q} \)
The magnitude of the vector \( \vec{P} - \vec{Q} \) is given by:
\[ |\vec{P} - \vec{Q}| = |2(\hat{j} - \hat{k})| = 2 \cdot |\hat{j} - \hat{k}| \]
To find \( |\hat{j} - \hat{k}| \):
\[ |\hat{j} - \hat{k}| = \sqrt{(0)^2 + (1)^2 + (-1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2} \]
Thus,
\[ |\vec{P} - \vec{Q}| = 2 \cdot \sqrt{2} \]

### Step 5: Find the unit vector
The unit vector \( \hat{z} \) in the direction of \( \vec{P} - \vec{Q} \) is given by:
\[ \hat{z} = \frac{\vec{P} - \vec{Q}}{|\vec{P} - \vec{Q}|} = \frac{2(\hat{j} - \hat{k})}{2\sqrt{2}} = \frac{\hat{j} - \hat{k}}{\sqrt{2}} \]

### Final Answer
Thus, the unit vector along \( \vec{P} - \vec{Q} \) is:
\[ \hat{z} = \frac{\hat{j} - \hat{k}}{\sqrt{2}} \]