Question

If $\left|\overrightarrow{a}\right|=4$ and $-3\le \lambda \le 2$, then write the range of $\left|\lambda \overrightarrow{a}\right|$.

Answer 1

It is given that

$$\begin{gathered} -3\le \lambda \le 2 \\ \Rightarrow -3\times \left|\overrightarrow{a}\right|\le \lambda \left|\overrightarrow{a}\right|\le 2\times \left|\overrightarrow{a}\right| \\ \Rightarrow -3\times 4\le \left|\lambda \overrightarrow{a}\right|\le 2\times 4\left(k\left|\overrightarrow{a}\right|=\left|k\overrightarrow{a}\right|,k\mathrm{is}\mathrm{scalar}\right) \\ \Rightarrow -12\le \left|\lambda \overrightarrow{a}\right|\le 8 \end{gathered}$$

Thus, the range of $\left|\lambda \overrightarrow{a}\right|$ is [−12, 8].