If the points P, Q, R with position vectors \(5\hat{\imath} + \lambda\hat{\jmath}\), \(20\hat{\imath} - \hat{\jmath}\) and \(15\hat{\imath} - 6\hat{\jmath}\) respectively are collinear, then the value of \(\lambda\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → −16
P, Q, R are collinear exactly when \(\overrightarrow{PQ}\) is parallel to \(\overrightarrow{QR}\).
\(\overrightarrow{QR} = (15 - 20)\hat{\imath} + (-6 + 1)\hat{\jmath} = -5\hat{\imath} - 5\hat{\jmath}\)
\(\overrightarrow{PQ} = (20 - 5)\hat{\imath} + (-1 - \lambda)\hat{\jmath} = 15\hat{\imath} + (-1 - \lambda)\hat{\jmath}\)
\(\overrightarrow{QR}\) has equal components, so the components of \(\overrightarrow{PQ}\) must also be equal:
\(-1 - \lambda = 15\)
\(\lambda = -16\)