2023HSCShort answer

Q11 (c)

2 marks

Question

Find a vector equation of the line through the points $A(-3,1,5)$ and $B(0,2,3)$ .

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Solution

Criteria	Marks
Provides a correct equation	2
Finds $\overrightarrow{AB}$ , or equivalent merit	1

Sample answer:

$\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}$
$=\overrightarrow{OB}+\overrightarrow{OA}$
$=\begin{pmatrix}0\\2\\3\end{pmatrix}-\begin{pmatrix}-3\\1\\5\end{pmatrix}=\begin{pmatrix}3\\1\\-2\end{pmatrix}$

$\therefore\text{ A vector equation of the line is }\vec r=\begin{pmatrix}-3\\1\\5\end{pmatrix}+\lambda\begin{pmatrix}3\\1\\-2\end{pmatrix},\quad\lambda\in\mathbb R$

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Syllabus alignment

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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