Let \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{w}\) be three vectors such that…
Let \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{w}\) be three vectors such that \(\mathbf{u}+\mathbf{v}+\mathbf{w}=0,|\mathbf{u}|=3,|\mathbf{v}|=5\) and \(|\mathbf{w}|=7\). Then the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is
\(60^{\circ}\)
\(70^{\circ}\)
\(80^{\circ}\)
\(90^{\circ}\)
Solution
Given that, \(|\mathbf{u}|=3,|\mathbf{v}|=5\) and \(|w|=7\)
and \(\mathbf{u}+\mathbf{v}+\mathbf{w}=0 \Rightarrow \mathbf{u}+\mathbf{v}=-\mathbf{w}\)
\(\begin{aligned}
& \Rightarrow \quad|\mathbf{u}|^2+|\mathbf{v}|^2+2 \mathbf{u}. \mathbf{v}=|\mathbf{w}|^2 \\
& \Rightarrow \quad 9+25+2 \times 3 \times 5 \cos \theta=49,
\end{aligned}\)
\{if angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(\theta\) \}
\(\Rightarrow \quad \cos \theta=\frac{1}{2} \Rightarrow \theta=60^{\circ}\)