If the points \( P(3,\mu,6),Q(-1,-1,2) \) and \( R(2,8,5) \) are collinear, then the value of 'μ' is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 11
P, Q and R are collinear when \( \overrightarrow{QP} \) is a scalar multiple of \( \overrightarrow{QR} \).
\( \displaystyle \overrightarrow{QR}=(2+1)\hat{\imath}+(8+1)\hat{\jmath}+(5-2)\hat{k} \)
\( \displaystyle =3\hat{\imath}+9\hat{\jmath}+3\hat{k} \)
\( \displaystyle \overrightarrow{QP}=(3+1)\hat{\imath}+(\mu+1)\hat{\jmath}+(6-2)\hat{k} \)
\( \displaystyle =4\hat{\imath}+(\mu+1)\hat{\jmath}+4\hat{k} \)
Comparing the x- and z-components, \( \overrightarrow{QP}=\frac{4}{3}\,\overrightarrow{QR} \). Then the y-components give
\( \displaystyle \mu+1=\frac{4}{3}\times 9 \)
\( \displaystyle \mu+1=12 \)
\( \displaystyle \mu=11 \)
Shortcut: Collinear points divide proportionally: from Q to R, x increases by 3 and y by 9, so y changes 3 times as fast as x. From Q to P, x increases by 4, so y must increase by 12: μ = −1 + 12 = 11.