2022HSCShort answer

Q11 (d)

3 marks

Question

A triangle is formed in three-dimensional space with vertices $A(1,-1,2)$ , $B(0,2,-1)$ and $C(2,1,1)$ . Find the size of $\angle ABC$ , giving your answer to the nearest degree.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Obtains correct dot product OR attempts to use the cosine rule using 3 correct lengths	2
Obtains one relevant vector OR obtains one relevant length	1

Sample answer:

A(1,-1,2),\quad B(0,2,-1),\quad C(2,1,1)

\mathbf a=\overrightarrow{BA}=\begin{pmatrix}1-0\\-1-2\\2-(-1)\end{pmatrix}=\begin{pmatrix}1\\-3\\3\end{pmatrix},\quad \mathbf b=\overrightarrow{BC}=\begin{pmatrix}2-0\\1-2\\1-(-1)\end{pmatrix}=\begin{pmatrix}2\\-1\\2\end{pmatrix}

\cos\angle ABC=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|} =\frac{2+3+6}{\sqrt{1+9+9}\,\sqrt{4+1+4}} =\frac{11}{3\sqrt{19}}

\therefore\ \angle ABC=\cos^{-1}\left(\frac{11}{3\sqrt{19}}\right)\approx33^\circ

Supplied marking guidance · human reviewed

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Syllabus alignment

Vectors

Further operations with three-dimensional vectorsUse vector magnitude, unit vectors and scalar products to solve problems and prove geometric results.

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