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

Given $|\vec{a}| = 5$ and $-2 \leq \lambda \leq 1$. The magnitude of $\lambda \vec{a}$ is calculated as: \[ |\lambda \vec{a}| = |\lambda| |\vec{a}| = |\lambda| \cdot 5 \] As $-2 \leq \lambda \leq 1$, the possible values of $|\lambda|$ are in the range \[0, 2\] (since $|\lambda| \geq 0$). Consequently, the range of $|\lambda \vec{a}|$ is: \[ [0 \cdot 5, 2 \cdot 5] = [0, 10] \] Therefore, the correct range for $|\lambda \vec{a}|$ is \[0, 10\].