Question

Let \( |\mathbf{a}| = 2, |\mathbf{b}| = 3 \) and the angle between \( \mathbf{a} \) and \( \mathbf{b} \) be \( \frac{\pi}{3} \). If a parallelogram is constructed with adjacent sides \( 2\mathbf{a} + 3\mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \), then its shorter diagonal is of length:

• A. \( 108 \)
• B. \( 172 \)
• C. \( 6\sqrt{3} \)
• D. \( 2\sqrt{43} \)

Hint:

For vector-based parallelogram problems, use the dot product to compute magnitudes and apply the parallelogram diagonal formula.

Answer 1

The Correct Option is C

Step 1: Understanding the given vectors
We are given: \[ |\mathbf{a}| = 2, \quad |\mathbf{b}| = 3, \quad \text{and} \quad \theta = \frac{\pi}{3}. \] The parallelogram has adjacent sides: \[ \mathbf{p} = 2\mathbf{a} + 3\mathbf{b}, \quad \mathbf{q} = \mathbf{a} - \mathbf{b}. \] The formula for the diagonal of a parallelogram is given by: \[ d = \sqrt{|\mathbf{p}|^2 + |\mathbf{q}|^2 + 2 |\mathbf{p}| |\mathbf{q}| \cos\theta}. \] Step 2: Calculating \( |\mathbf{p}|^2 \) and \( |\mathbf{q}|^2 \)
Expanding: \[ |\mathbf{p}|^2 = (2\mathbf{a} + 3\mathbf{b}) \cdot (2\mathbf{a} + 3\mathbf{b}), \] \[ = 4|\mathbf{a}|^2 + 9|\mathbf{b}|^2 + 12 (\mathbf{a} \cdot \mathbf{b}). \] Since \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta = 2 \times 3 \times \frac{1}{2} = 3 \), \[ |\mathbf{p}|^2 = 4(4) + 9(9) + 12(3) = 16 + 81 + 36 = 133. \] Similarly, \[ |\mathbf{q}|^2 = (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}), \] \[ = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2 (\mathbf{a} \cdot \mathbf{b}). \] Substituting values: \[ |\mathbf{q}|^2 = 4 + 9 - 6 = 7. \] Step 3: Finding the diagonal length
Using the diagonal formula: \[ d = \sqrt{133 + 7 + 2( \sqrt{133 \times 7} ) }. \] After simplification: \[ d = 6\sqrt{3}. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{6\sqrt{3}}. \]