Let \(\vec{a}\) and \(\vec{b}\) be two non-collinear vectors. If the vectors \((\lambda-2)\vec{a}+\vec{b}\) and \((2\lambda+1)\vec{a}-\vec{b}\) are collinear, then the value of \(\lambda\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{3}\)
The vector \((2\lambda+1)\vec{a}-\vec{b}\) is non-zero, because \(\vec{a}\) and \(\vec{b}\) are non-collinear. So the two vectors are collinear exactly when, for some scalar t,
\((\lambda-2)\vec{a}+\vec{b} = t\left[(2\lambda+1)\vec{a}-\vec{b}\right]\)
Since \(\vec{a}\) and \(\vec{b}\) are non-collinear, the coefficients of \(\vec{a}\) and of \(\vec{b}\) on the two sides must be equal:
\(1 = -t\), so \(t = -1\)
\(\lambda-2 = -(2\lambda+1)\)
\(3\lambda = 1\)
\( \displaystyle \lambda = \frac{1}{3} \)