Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are unit vectors, then which of the following values of $\overrightarrow{a}.\overrightarrow{b}$is not possible?
(a) $\sqrt{3}$

(b) $\sqrt{3}/2$

(c) $1/\sqrt{2}$

(d) −1/2

Answer 1

(a) $\sqrt{3}$

$$\begin{gathered} \text{It is given that}\overrightarrow{a}\text{and}\overrightarrow{b}\text{are unit vectors.} \\ \Rightarrow \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=1 \\ \text{Now,} \\ \overrightarrow{a}.\overrightarrow{b} \\ =\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ =\left(1\right)\left(1\right)\cos \theta \\ =\cos \theta \\ \text{The range of}\cos \theta \text{is}\left[-1,1\right]\text{.} \\ \therefore \sqrt{\text{3}}\text{is not a possible value of}\cos \theta \mathrm{as}\mathrm{it}\mathrm{is}\mathrm{greater}\mathrm{than}1\text{.} \end{gathered}$$