If \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors such that \(\mathbf{a}+\mathbf{b}\) is also a unit…
If \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors such that \(\mathbf{a}+\mathbf{b}\) is also a unit vector, then the angle between a and \(\mathbf{b}\) is ____
\(60^{\circ}\)
\(90^{\circ}\)
\(30^{\circ}\)
\(120^{\circ}\)
Solution
For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), it is given \(\mathbf{a}+\mathbf{b}\) is also unit vector, so
\(\begin{aligned}
& |\mathbf{a}+\mathbf{b}|=\mathbf{1} \Rightarrow(\mathbf{a}+\mathbf{b})^2=1 \\
& \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2 \mathbf{a} \cdot \mathbf{b}=\mathbf{l} \quad \Rightarrow \quad \mathbf{1}+\mathbf{1}+2 \mathbf{a} \cdot \mathbf{b}=\mathbf{1} \\
& \Rightarrow \quad \mathbf{a} \cdot \mathbf{b}=-\frac{1}{2} \\
\end{aligned}\)
Let angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\boldsymbol{\theta}\)
Then \(\cos \theta=-\frac{1}{2} \Rightarrow \theta=120^{\circ}\)
Hence, option (d) is correct.