Question

$\mathrm{If}\left|\overrightarrow{a}\right|=3\mathrm{and}-1\le \lambda \le 2,\mathrm{then}\left|\lambda \overrightarrow{a}\right|$ lies in the interval
(a) [0, 6]
(b) [-3, 6]
(c) [3,6]
(d) [1, 2]

Answer 1

Given: $\left|\overrightarrow{a}\right|=3\mathrm{and}-1\le \lambda \le 2$

$$\begin{gathered} \left|\lambda \overrightarrow{a}\right|=\lambda \left|\overrightarrow{a}\right| \\ \mathrm{since},-1\le \lambda \le 2 \\ \Rightarrow -1\left|\overrightarrow{a}\right|\le \lambda \left|\overrightarrow{a}\right|\le 2\left|\overrightarrow{a}\right| \\ \Rightarrow -1\times 3\le \lambda \left|\overrightarrow{a}\right|\le 2\times 3\left(\because \left|\overrightarrow{a}\right|=3\right) \\ \Rightarrow -3\le \left|\lambda \overrightarrow{a}\right|\le 6 \\ \Rightarrow \left|\lambda \overrightarrow{a}\right|\in \left[-3,6\right] \end{gathered}$$

Hence, the correct option is (b).