<p><strong>Ex. 64:</strong> Find the least positive integral value of x for which the angle between vectors \(\vec{a} = x\vec{i} - 3\vec{j} - \vec{k}\) and \(\vec{b} = 2x\vec{i} + x\vec{j} - \vec{k}\) is acute.</p>
Step-by-Step Solution
Key Concept: For an acute angle between vectors, use the condition that \(|\vec{a} + \vec{b}| > |\vec{a} - \vec{b}|\), which is equivalent to \(\vec{a} \cdot \vec{b} > 0\).
Step 1: For the angle to be acute, we need \(|\vec{a} + \vec{b}| > |\vec{a} - \vec{b}|\). Step 2: Calculate \(\vec{a} + \vec{b} = 3x\hat{i} + (x-3)\hat{j} - 2\hat{k}\) and \(\vec{a} - \vec{b} = -x\hat{i} - (x+3)\hat{j}\). Step 3: Squaring both sides: \[9x^2 + (x-3)^2 + 4 > x^2 + (x+3)^2\] \[9x^2 + x^2 - 6x + 9 + 4 > x^2 + x^2 + 6x + 9\] \[8x^2 - 12x + 4 > 0\] \[2x^2 - 3x + 1 > 0\] Step 4: Factor: \((2x - 1)(x - 1) > 0\). \[x < \frac{1}{2} \text{ or } x > 1\] Step 5: The least positive integral value is \(x = 2\).
Correct Answer: 2