2022HSCShort answer

Q14 (a) (iv)

2 marks

Question

The point $P$ is defined by
$\overrightarrow{AP}=-6\overrightarrow{AB}-8\overrightarrow{AC}.$

Does $P$ lie on the line $AL$ ? Justify your answer.

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Solution

Criteria	Marks
Provides correct solution	2
Finds the vector $\overrightarrow{AL}$ , or equivalent merit	1

Sample answer:
Write both $\overrightarrow{AP}$ and $\overrightarrow{AL}$ in terms of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

\overrightarrow{AL}=\overrightarrow{AB}+\overrightarrow{BL} =\overrightarrow{AB}+\frac47\overrightarrow{BC} =\overrightarrow{AB}+\frac47(\overrightarrow{BA}+\overrightarrow{AC}) =\frac37\overrightarrow{AB}+\frac47\overrightarrow{AC}

$\overrightarrow{AP}=-6\overrightarrow{AB}-8\overrightarrow{AC}=-14\left(\frac37\overrightarrow{AB}+\frac47\overrightarrow{AC}\right)=-14\overrightarrow{AL}$
$\overrightarrow{AP}=-14\overrightarrow{AL}$ , so $\overrightarrow{AP}$ and $\overrightarrow{AL}$ are parallel, and $A$ is common to those vectors, so $P$ lies on the line $AL$ .

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Vectors

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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