2022HSCMultiple choice

Q3

1 mark

Question

Let $A$ , $B$ , $P$ be three points in three-dimensional space with $A\ne B$ .

If $P$ is on the line $AB$ , then there exists a real number $\lambda$ such that $\overrightarrow{AP}=\lambda\overrightarrow{AB}$ .

Which of the following is the contrapositive of this statement?

A. If for all real numbers $\lambda$ , $\overrightarrow{AP}=\lambda\overrightarrow{AB}$ , then $P$ is on the line $AB$ .
B. If for all real numbers $\lambda$ , $\overrightarrow{AP}\ne\lambda\overrightarrow{AB}$ , then $P$ is not on the line $AB$ .
C. If there exists a real number $\lambda$ such that $\overrightarrow{AP}=\lambda\overrightarrow{AB}$ , then $P$ is on the line $AB$ .
D. If there exists a real number $\lambda$ such that $\overrightarrow{AP}\ne\lambda\overrightarrow{AB}$ , then $P$ is not on the line $AB$ .

Try the question before revealing the answer.

Syllabus alignment

ProofVectors

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.
Vectors and vector equations of linesUse coordinates and vector equations for curves, lines, line segments and spheres in two and three dimensions.

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