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Answer the following.
If a⃗ and b⃗ are non-collinear vectors, find the value of x such that the vectors α⃗ = (x - 2)a⃗ +…
If $\vec{a}$ and $\vec{b}$ are non-collinear vectors, find the value of $x$ such that the vectors $\vec{\alpha} = (x - 2)\vec{a} + \vec{b}$ and $\vec{\beta} = (3 + 2x)\vec{a} - 2\vec{b}$ are collinear.
Answer
Answer
$\vec\alpha$ and $\vec\beta$ are collinear, so $\vec\alpha = k\vec\beta$ for some scalar $k$.
$(x-2)\vec a+\vec b = k(3+2x)\vec a - 2k\vec b$
Since $\vec a,\vec b$ are non-collinear, compare coefficients: $x-2 = k(3+2x)$ and $1 = -2k$.
$k = -\frac12$, so $x-2 = -\frac12(3+2x) = -\frac32 - x$.
$2x = \frac12$
Hence $x = \frac14$.
Final answer: $\frac{1}{4}$