Question:

<div class="solution"> <p><strong>Step 1:</strong> Given the equation \(\vec{a} = \vec{b} \times \vec{c} + 2\vec{b}\), we need to find the sum of possible values of \(|2\vec{a} + \vec{b} + \vec{c}|\). First, let's express \(2\vec{a} + \vec{b} + \vec{c}\) in terms of \(\vec{b}\) and \(\vec{c}\) using the given relation.</p> <p><strong>Step 2:</strong> Substitute \(\vec{a}\) from the given equation into \(2\vec{a} + \vec{b} + \vec{c}\) to get \(2(\vec{b} \times \vec{c} + 2\vec{b}) + \vec{b} + \vec{c}\). Simplifying this yields \(2\vec{b} \times \vec{c} + 4\vec{b} + \vec{b} + \vec{c} = 2\vec{b} \times \vec{c} + 5\vec{b} + \vec{c}\).</p> <p><strong>Step 3:</strong> To find the magnitude of \(2\vec{a} + \vec{b} + \vec{c}\), we use the expression \(|2\vec{a} + \vec{b} + \vec{c}| = |2\vec{b} \times \vec{c} + 5\vec{b} + \vec{c}|\). Since \(\vec{b} \times \vec{c}\) is orthogonal to both \(\vec{b}\) and \(\vec{c}\), we can use the Pythagorean theorem to find the magnitude.</p> <p><strong>Step 4:</strong> The magnitude \(|2\vec{b} \times \vec{c}|\) can be found using the property of the cross product: \(|\vec{b} \times \vec{c}| = |\vec{b}||\vec{c}|\sin(\theta)\), where \(\theta\) is the angle between \(\vec{b}\) and \(\vec{c}\). Given \(|\vec{b}| = |\vec{c}| = 2\), we have \(|2\vec{b} \times \vec{c}| = 2 \cdot 2 \cdot 2 \cdot \sin(\theta) = 8\sin(\theta)\).</p> <p><strong>Step 5:</strong> The magnitude of \(5\vec{b} + \vec{c}\) can be found using \(|5\vec{b} + \vec{c}| = \sqrt{(5|\vec{b}|)^2 + |\vec{c}|^2 + 2 \cdot 5|\vec{b}| \cdot |\vec{c}| \cdot \cos(\theta)}\), which simplifies to \(\sqrt{25 \cdot 4 + 4 + 20 \cdot 2 \cdot \cos(\theta)} = \sqrt{100 + 40\cos(\theta)}\).</p> <p><strong>Step 6:</strong> Applying the Pythagorean theorem, \(|2\vec{a} + \vec{b} + \vec{c}| = \sqrt{|2\vec{b} \times \vec{c}|^2 + |5\vec{b} + \vec{c}|^2} = \sqrt{(8\sin(\theta))^2 + (100 + 40\cos(\theta))}\). Simplifying this yields \(\sqrt{64\sin^2(\theta) + 100 + 40\cos(\theta)}\).</p> <p><strong>Step 7:</strong> Using the trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), we can rewrite the expression in terms of \(\cos(\theta)\) only: \(\sqrt{64(1 - \cos^2(\theta)) + 100 + 40\cos(\theta)} = \sqrt{164 - 64\cos^2(\theta) + 40\cos(\theta)}\).</p> <p><strong>Step 8:</strong> To find the maximum and minimum values of \(|2\vec{a} + \vec{b} + \vec{c}|\), we need to find the critical points of the function \(f(\cos(\theta))