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Question

Prove that:
(2^i+3^j+5^k),(^i+2^j+3^k) and (7^i^k) are collinear.

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Solution

Let A=2^i+3^j+5^k, B=^i+2^j+3^k and C=7^i^k.
Now, line AB=A+λ(BA)
AB=2^i+3^j+5^k+λ(^i+2^j+3^k(2^i+3^j+5^k))
AB=2^i+3^j+5^k+λ(3^i^j2^k)
Now, if C lies on AB, we should have
7^i^k=2^i+3^j+5^k+λ(3^i^j2^k)
If λ=3, AB=7^i^k
OR
Let A=2^i+3^j+5^k, B=^i+2^j+3^k and C=7^i^k.
Now, line AB=BA
AB=^i+2^j+3^k(2^i+3^j+5^k)
AB=3^i^j2^k
Now, AC=CA
AC=7^i^k(2^i+3^j+5^k)
AC=9^i3^j6^k
Also, if AB and AC are collinear, AB×AC should be 0.
AB×AC=∣ ∣ ∣^i^j^k312936∣ ∣ ∣
=(66)^i(18+18)^j+(9+9)^k
=0

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