If a and b are non-collinear vectors, then the value of for which the vectors u=(-2) a+ b …

Question

If $\vec a$ and $\vec b$ are non-collinear vectors, then the value of $\alpha$ for which the vectors $\vec u=(\alpha-2)\vec a+\vec b$ and $\vec v=(2+3\alpha)\vec a-3\vec b$ are collinear is:

Accepted Answer

For collinearity, $k\vec u+\vec v=\vec 0$ for some scalar $k$: $[k(\alpha-2)+2+3\alpha]\vec a+(k-3)\vec b=\vec 0$.

Since $\vec a,\vec b$ are non-collinear (independent): $k-3=0\implies k=3$, and

$$3(\alpha-2)+2+3\alpha=0 \implies 6\alpha-4=0 \implies \alpha=\frac23$$

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