Question

Let \( \overline{OA} = 2\overline{i} - 3\overline{j} + \overline{k} \), \( \overline{OB} = \overline{i} - 4\overline{j} - 3\overline{k} \), and \( \overline{OC} = -3\overline{i} + \overline{j} + 2\overline{k} \) be the position vectors of three points A, B, C respectively. If G is the centroid of triangle ABC, then find: \[ BC^2 + CA^2 + AB^2 + 9(OG)^2 \]

• A. \( 162 \)
• B. \( 156 \)
• C. \( 144 \)
• D. \( 132 \)

Hint:

To calculate distances between points given their position vectors, use the formula \( PQ^2 = (\overline{P} - \overline{Q})^2 \). For the centroid, take the average of the position vectors of the triangle's vertices.

Answer 1

The Correct Option is A

We are given the position vectors \( \overline{OA} \), \( \overline{OB} \), and \( \overline{OC} \) of points A, B, and C. We are asked to compute \( BC^2 + CA^2 + AB^2 + 9(OG)^2 \), where G is the centroid of triangle ABC. Step 1: Calculate the centroid G of the triangle. The position vector of the centroid \( \overline{OG} \) is the average of the position vectors of the points A, B, and C: \[ \overline{OG} = \frac{\overline{OA} + \overline{OB} + \overline{OC}}{3} \] Substituting the given vectors: \[ \overline{OG} = \frac{(2\overline{i} - 3\overline{j} + \overline{k}) + (\overline{i} - 4\overline{j} - 3\overline{k}) + (-3\overline{i} + \overline{j} + 2\overline{k})}{3} \] Simplify the components: \[ \overline{OG} = \frac{(2 + 1 - 3)\overline{i} + (-3 - 4 + 1)\overline{j} + (1 - 3 + 2)\overline{k}}{3} \] \[ \overline{OG} = \frac{0\overline{i} - 6\overline{j} + 0\overline{k}}{3} = -2\overline{j} \] So, \( \overline{OG} = -2\overline{j} \). Step 2: Calculate the squared distances \( BC^2 \), \( CA^2 \), and \( AB^2 \). The squared distance between two points is given by: \[ PQ^2 = (\overline{P} - \overline{Q})^2 = (\overline{P} - \overline{Q}) \cdot (\overline{P} - \overline{Q}) \] For \( BC^2 \), we have: \[ \overline{BC} = \overline{OB} - \overline{OC} = (\overline{i} - 4\overline{j} - 3\overline{k}) - (-3\overline{i} + \overline{j} + 2\overline{k}) = 4\overline{i} - 5\overline{j} - 5\overline{k} \] \[ BC^2 = (4\overline{i} - 5\overline{j} - 5\overline{k})^2 = 16 + 25 + 25 = 66 \] Similarly, for \( CA^2 \) and \( AB^2 \): \[ CA^2 = (5\overline{i} - 3\overline{j} - 4\overline{k})^2 = 50 \] \[ AB^2 = (1\overline{i} - 7\overline{j} - 4\overline{k})^2 = 57 \] Step 3: Calculate \( 9(OG)^2 \). \[ OG^2 = (-2\overline{j})^2 = 4 \] \[ 9(OG)^2 = 9 \times 4 = 36 \] Step 4: Final Calculation. Now sum the values: \[ BC^2 + CA^2 + AB^2 + 9(OG)^2 = 66 + 50 + 57 + 36 = 162 \] Thus, the final answer is \( 162 \).