`P(0, 3, -2), Q(3, 7, -1) and R(1, -3, -1)` are 3 given points. Find `vec (PQ)`

Answer

To find the vector \(\vec{PQ}\) from point \(P(0, 3, -2)\) to point \(Q(3, 7, -1)\), we can follow these steps:

### Step 1: Identify the coordinates of points P and Q
- Point \(P\) has coordinates \((x_1, y_1, z_1) = (0, 3, -2)\)
- Point \(Q\) has coordinates \((x_2, y_2, z_2) = (3, 7, -1)\)

### Step 2: Use the formula for the vector \(\vec{PQ}\)
The vector \(\vec{PQ}\) can be calculated using the formula:
\[ \vec{PQ} = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} + (z_2 - z_1) \hat{k} \]

### Step 3: Substitute the coordinates into the formula
Now, we will substitute the coordinates of points \(P\) and \(Q\) into the formula:
- \(x_2 - x_1 = 3 - 0 = 3\)
- \(y_2 - y_1 = 7 - 3 = 4\)
- \(z_2 - z_1 = -1 - (-2) = -1 + 2 = 1\)

### Step 4: Write the vector \(\vec{PQ}\)
Now, substituting these values into the vector formula:
\[ \vec{PQ} = 3 \hat{i} + 4 \hat{j} + 1 \hat{k} \]

### Step 5: Final answer
Thus, the vector \(\vec{PQ}\) is:
\[ \vec{PQ} = 3 \hat{i} + 4 \hat{j} + 1 \hat{k} \]

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