Mathematics - Vector Question with Solution | TestHub

Since, $\vec{u}$ and $\vec{v}$ are collinear, therefore $$\begin{array}{rl}& \vec{u}+\vec{v}=0\\ \Rightarrow & [k(\alpha -2)+2+3\alpha ]\vec{a}+(k-3)\vec{b}=0\end{array}$$ Since $\vec{a}$ and $\vec{b}$ are non-collinear, then for some constant $m$ and $n$, $$m\vec{a}+n\vec{b}=0\Rightarrow m=0,n=0$$ Hence from equation (i) $$\begin{array}{rl}& k-3=0\Rightarrow k=3\\ & \text{ And }k(\alpha -2)+2+3\alpha =0\\ & \Rightarrow 3(\alpha -2)+2+3\alpha =0\Rightarrow \alpha =\frac{2}{3}\end{array}$$