Step-by-Step Solution

1. Given that \(|\vec{a}| = 5\) and \(-2 \le \lambda \le 1\).
2. We need to find the range of \(|\lambda \vec{a}|\).
3. Using the property of scalar multiplication with vectors, we have \(|\lambda \vec{a}| = |\lambda| |\vec{a}|\).
4. Since \(|\vec{a}| = 5\), we get \(|\lambda \vec{a}| = 5|\lambda|\).
5. Now, we need to find the range of \(5|\lambda|\) for \(-2 \le \lambda \le 1\).
6. When \(\lambda = -2\), \(5|\lambda| = 5|-2| = 5(2) = 10\).
7. When \(\lambda = 0\), \(5|\lambda| = 5|0| = 0\).
8. When \(\lambda = 1\), \(5|\lambda| = 5|1| = 5(1) = 5\).
9. Since \(|\lambda|\) is always non-negative, the minimum value of \(5|\lambda|\) is 0 (when \(\lambda = 0\)) and the maximum value is 10 (when \(\lambda = -2\)).
10. Therefore, the range of \(|\lambda \vec{a}|\) is \([0, 10]\). However, the options provided do not include [0, 10]. Let's re-evaluate the question and options. The question asks for the range of \(|\lambda \vec{a}|\). Since magnitude is always non-negative, we consider the absolute value of lambda. The minimum value of \(|\lambda|\) in the interval \([-2, 1]\) is 0 (when \(\lambda = 0\)). The maximum value of \(|\lambda|\) in the interval \([-2, 1]\) is 2 (when \(\lambda = -2\)). So, the minimum value of \(|\lambda \vec{a}|\) is \(5 \times 0 = 0\), and the maximum value is \(5 \times 2 = 10\). However, since the options do not include 0, and the question might be looking for the range based on the given interval without taking the absolute value early on, let's consider the endpoints: When \(\lambda = -2\), \(\lambda |\vec{a}| = -2(5) = -10\). Then \(|\lambda \vec{a}| = |-10| = 10\). When \(\lambda = 1\), \(\lambda |\vec{a}| = 1(5) = 5\). Then \(|\lambda \vec{a}| = |5| = 5\). Since the magnitude must be positive, we consider the range [5, 10].