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Question

Show that the following poionts are collinear
(i) A(3,2,4), B(9,8,10), C(2,3,1)
(ii) P(4,5,2), Q(3,2,4), R(5,8,0)

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Solution

(i)A(3,2,4),B(9,8,10),C(2,3,1)

For A,B,C to be collinear,

λa+μb+δc=0 where λ,μ,δ0

(3^i+2^j4^k)λ+(9^i+8^j10^k)μ+(2^i3^j+^k)δ=0(3λ+9μ2δ)^i+(2λ+8μ3δ)^j+(4λ10μ+1δ)^k=03λ+9μ2δ=02λ+8μ3δ=04λ10μ+1δ=0

Solving by cramer's Rule,

∣ ∣3922834101∣ ∣=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.

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