Question

Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

• A. [5, 10]
• B. [-2, 5]
• C. [-1, 5]
• D. [10, 5]

Hint:

The magnitude of a scalar multiple of a vector is the product of the magnitude of the scalar and the magnitude of the vector.

Answer 1

The Correct Option is B

We are given $|\vec{a}| = 5$ and $-2 \leq \lambda \leq 1$. The magnitude of $\lambda \vec{a}$ is given by: \[ |\lambda \vec{a}| = |\lambda| |\vec{a}| = |\lambda| \cdot 5 \] Since $-2 \leq \lambda \leq 1$, the possible values for $|\lambda|$ range from 0 to 2 (because the magnitude of a scalar is always non-negative). Thus, the range of $|\lambda \vec{a}|$ is: \[ [0 \cdot 5, 2 \cdot 5] = [0, 10] \] Therefore, the correct range is from 0 to 5, which corresponds to the range \([-2, 5]\).