Question

Let \(\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}\), \(\vec{b} = 6\hat{i} + 3\hat{j} + 3\hat{k}\). If \(2\vec{a} + 3\vec{b}\) and \(\vec{a} - \vec{b}\) are two adjacent sides of a triangle then square of area of the triangle is

• A. 1800
• B. 1600
• C. 2000
• D. 2200

Hint:

To simplify \((m\vec{a} + n\vec{b}) \times (p\vec{a} + q\vec{b})\), use the scalar property: the result is always \((mq - np)(\vec{a} \times \vec{b})\). This saves time over calculating each vector side individually.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The area of a triangle with adjacent sides represented by vectors \(\vec{u}\) and \(\vec{v}\) is given by \( \text{Area} = \frac{1}{2} |\vec{u} \times \vec{v}| \). We first calculate the resultant vectors for the sides and then find their cross product.

Step 2: Key Formula or Approach:
Let \(\vec{u} = 2\vec{a} + 3\vec{b}\) and \(\vec{v} = \vec{a} - \vec{b}\). Area \(A = \frac{1}{2} |(2\vec{a} + 3\vec{b}) \times (\vec{a} - \vec{b})| \). Using properties of cross products: \(\vec{a} \times \vec{a} = 0\) and \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\).

Step 3: Detailed Explanation:
1. Simplify the cross product: \[ \vec{u} \times \vec{v} = (2\vec{a} + 3\vec{b}) \times (\vec{a} - \vec{b}) \] \[ = 2(\vec{a} \times \vec{a}) - 2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{a}) - 3(\vec{b} \times \vec{b}) \] \[ = 0 - 2(\vec{a} \times \vec{b}) - 3(\vec{a} \times \vec{b}) - 0 = -5(\vec{a} \times \vec{b}) \] 2. Calculate \(\vec{a} \times \vec{b}\): \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & 3 & 3
6 & 3 & 3 \end{vmatrix} = \hat{i}(9-9) - \hat{j}(6-18) + \hat{k}(6-18) = 12\hat{j} - 12\hat{k} \] 3. Find the magnitude: \[ |\vec{u} \times \vec{v}| = |-5(12\hat{j} - 12\hat{k})| = 60 |\hat{j} - \hat{k}| = 60\sqrt{1^2 + (-1)^2} = 60\sqrt{2} \] 4. Area and Square of Area: \[ \text{Area} = \frac{1}{2} (60\sqrt{2}) = 30\sqrt{2} \] \[ \text{Area}^2 = (30\sqrt{2})^2 = 900 \times 2 = 1800 \]

Step 4: Final Answer:
The square of the area of the triangle is 1800.