Find a unit vector in the direction from point `P(1,-1,2)` to point `Q(-1,1,1)`.

Answer

To find a unit vector in the direction from point \( P(1, -1, 2) \) to point \( Q(-1, 1, 1) \), we will follow these steps:

### Step 1: Find the vector \( \vec{PQ} \)

The vector \( \vec{PQ} \) can be calculated using the formula:
\[ \vec{PQ} = \vec{Q} - \vec{P} \]
Where \( \vec{P} = (1, -1, 2) \) and \( \vec{Q} = (-1, 1, 1) \).

Calculating the components:
\[ \vec{PQ} = (-1 - 1, 1 - (-1), 1 - 2) = (-2, 2, -1) \]
Thus, \( \vec{PQ} = -2 \hat{i} + 2 \hat{j} - 1 \hat{k} \).

### Step 2: Calculate the magnitude of \( \vec{PQ} \)

The magnitude \( |\vec{PQ}| \) is given by:
\[ |\vec{PQ}| = \sqrt{(-2)^2 + (2)^2 + (-1)^2} \]
Calculating the squares:
\[ |\vec{PQ}| = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 3: Find the unit vector in the direction of \( \vec{PQ} \)

The unit vector \( \hat{u} \) in the direction of \( \vec{PQ} \) is given by:
\[ \hat{u} = \frac{\vec{PQ}}{|\vec{PQ}|} \]
Substituting the values we found:
\[ \hat{u} = \frac{-2 \hat{i} + 2 \hat{j} - 1 \hat{k}}{3} \]
Thus, the unit vector is:
\[ \hat{u} = -\frac{2}{3} \hat{i} + \frac{2}{3} \hat{j} - \frac{1}{3} \hat{k} \]

### Final Answer
The unit vector in the direction from point \( P \) to point \( Q \) is:
\[ \hat{u} = -\frac{2}{3} \hat{i} + \frac{2}{3} \hat{j} - \frac{1}{3} \hat{k} \]

---