(i)A(3,2,−4),B(9,8,−10),C(−2,−3,1)
∴ For A,B,C to be collinear,
λ→a+μ→b+δ→c=0 where λ,μ,δ≠0
∴(3^i+2^j−4^k)λ+(9^i+8^j−10^k)μ+(−2^i−3^j+^k)δ=0(3λ+9μ−2δ)^i+(2λ+8μ−3δ)^j+(−4λ−10μ+1δ)^k=0∴3λ+9μ−2δ=02λ+8μ−3δ=0−4λ−10μ+1δ=0
Solving by cramer's Rule,
∣∣
∣∣39−228−3−4−101∣∣
∣∣=0
∴ unique solutions for μ,λ,δ exist and scalars are non zero.
∴ A,B,C are collinear.
(ii)P(4,5,2),Q(3,2,4),R(5,8,0)
Similarly,
∣∣
∣∣452324580∣∣
∣∣=0
unique solutions exists and all the scalars are non zero.
∴ P,Q,R are collinear.