Question

If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) will be

• A. \( -3\hat{i} - 5\hat{j} + 4\hat{k} \)
• B. \( 3\hat{i} + 5\hat{j} + 4\hat{k} \)
• C. \( -3\hat{i} - 5\hat{j} - 4\hat{k} \)
• D. \( 3\hat{i} + 5\hat{j} - 4\hat{k} \)

Hint:

A common mistake is to subtract in the wrong order (P - Q instead of Q - P). Always remember the rule: Terminal minus Initial. For \( \vec{PQ} \), Q is the terminal point and P is the initial point.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the vector connecting two points, say from point P to point Q, we subtract the position vector of the initial point (P) from the position vector of the terminal point (Q).
Step 2: Key Formula or Approach:
If P has coordinates \((x_1, y_1, z_1)\) and Q has coordinates \((x_2, y_2, z_2)\), the vector \( \vec{PQ} \) is given by: \[ \vec{PQ} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} \] This can also be written as \( \vec{PQ} = \text{Position Vector of Q} - \text{Position Vector of P} \).
Step 3: Detailed Explanation:
We are given the coordinates of the points:
P = (2, 3, 0), so \(x_1 = 2, y_1 = 3, z_1 = 0\).
Q = (–1, –2, –4), so \(x_2 = -1, y_2 = -2, z_2 = -4\).
Now, we apply the formula: \[ \vec{PQ} = (-1 - 2)\hat{i} + (-2 - 3)\hat{j} + (-4 - 0)\hat{k} \] \[ \vec{PQ} = (-3)\hat{i} + (-5)\hat{j} + (-4)\hat{k} \] \[ \vec{PQ} = -3\hat{i} - 5\hat{j} - 4\hat{k} \] Step 4: Final Answer:
The vector \( \vec{PQ} \) is \( -3\hat{i} - 5\hat{j} - 4\hat{k} \). This matches option (C).