EXPLANATION.

Points A B C are the position vectors,

\sf \implies (A) = 3 \hat {i} - y \hat {j} + 2 \hat{k}.⟹(A)=3i^−yj^+2k^.

\sf \implies (B) = 5 \hat {i} - \hat {j} + \hat {k}.⟹(B)=5i^−j^+k^.

\sf \implies ( C) =3x \hat {i} + 3 \hat{j} - \hat {k}.⟹(C)=3xi^+3j^−k^.

\sf \implies \vec {AB} = (5 \hat {i} - \hat {j} + \hat {k} ) - [ 3 \hat {i} - y \hat {j} + 2 \hat {k}] .⟹AB=(5i^−j^+k^)−[3i^−yj^+2k^].

\sf \implies \vec {AB} = 5 \hat {i} - \hat {j} + \hat {k} - 3 \hat {i} + y \hat {j} - 2 \hat {k}⟹AB=5i^−j^+k^−3i^+yj^−2k^

\sf \implies \vec {AB} = 2 \hat {i} - (1 - y ) \hat {j} - 2 \hat {k} .⟹AB=2i^−(1−y)j^−2k^.

\sf \implies \vec {BC} = 3X \hat {i} + 3 \hat {j} - \hat {k} - [ 5 \hat {i} - \hat {j} + \hat {k} ] .⟹BC=3Xi^+3j^−k^−[5i^−j^+k^].

\sf \implies \vec {BC} = 3x \hat {i} + 3 \hat {j} - \hat {j} - 5 \hat {i} + \hat {j} - \hat {k} .⟹BC=3xi^+3j^−j^−5i^+j^−k^.

\sf \implies \vec {BC} = (3x - 5) \hat {i} + 4 \hat {j} - 2 \hat {k} .⟹BC=(3x−5)i^+4j^−2k^.

\sf \implies as \ we\ know \ that = \vec {r} = \vec {a} + \lambda \vec {b} .⟹as we know that=r=a+λb.

\sf \implies 2 \hat {i} - (1 - y) \hat {j} - 2 \hat {k} =\lambda [(3x - 5) \hat{i} + 4 \hat {j} - 2 \hat {k} ] .⟹2i^−(1−y)j^−2k^=λ[(3x−5)i^+4j^−2k^].

\sf \implies 2 = \lambda(3x - 5). =(1).⟹2=λ(3x−5).=(1).

\sf \implies -(1 - y) = 4 \lambda =(2).⟹−(1−y)=4λ=(2).

\sf \implies -2 = -2 \lambda = (3)⟹−2=−2λ=(3)

From equation, (3) we get.

⇒ 2 = 2λ.

⇒ λ = 1.

Put the value of λ in equation, (1) and (2) we get.

⇒ 2 = 1(3x - 5).

⇒ 2 = 3x - 5.

⇒ 8 = 3x.

⇒ x = 8/3.

⇒ -(1 - y) = 4.

⇒ - 1 + y = 4.

⇒ y = 5.

Value of x = 8/3   y = 5  λ = 1

As the point divides in the ratio of k or 1, we get,

Co-ordinates are,

A = (3,-1,2).

B = (5,-1,1).

C = (8,3,-1).

\sf \implies \vec {OA} = 3 \hat {i} - y \hat {j} + 2\hat {k}

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