Let A, B, P be three points in three-dimensional space with A ≠ B - HSC - SSCE Mathematics Extension 2 - Question 3 - 2022 - Paper 1

The contrapositive of a statement of the form "If P, then Q" is "If not Q, then not P."

In this case, the conclusion Q is: "There exists a real number λ such that \vec{AP} = λ \vec{AB}." The negation (not Q) would be: "For all real numbers λ, \vec{AP} ≠ λ \vec{AB}."

The premise P is: "P is on the line AB." The negation (not P) is: "P is not on the line AB."

Therefore, the contrapositive of the original statement is: "If for all real numbers λ, \vec{AP} ≠ λ \vec{AB}, then P is not on the line AB."

The correct option corresponding to the contrapositive is B: "If for all real numbers λ, \vec{AP} ≠ λ \vec{AB}, then P is not on the line AB."