Let A, B, P be three points in three-dimensional space with A ≠ B - HSC - SSCE Mathematics Extension 2 - Question 3 - 2022 - Paper 1
Question 3
Let A, B, P be three points in three-dimensional space with A ≠ B.
Consider the following statement.
If P is on the line AB, then there exists a real number λ such... show full transcript
Worked Solution & Example Answer:Let A, B, P be three points in three-dimensional space with A ≠ B - HSC - SSCE Mathematics Extension 2 - Question 3 - 2022 - Paper 1
Step 1
Identify the Original Statement
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Answer
The original statement can be expressed as:
If P is on the line AB, then there exists a real number λ such that
ar{AP} = λar{AB}.
Step 2
Formulate the Contrapositive
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Answer
The contrapositive of a statement of the form "If A, then B" is "If not B, then not A." Here, the contrapositive will be:
If there does not exist a real number λ such that
ar{AP} = λar{AB}, then P is not on the line AB.
Step 3
Select the Correct Option
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Answer
From the options provided, the correct choice that matches the contrapositive is:
B. If for all real numbers λ, ar{AP} ≠ λar{AB}, then P is not on the line AB.
