2021HSCShort answer

Q11 (c)

3 marks

Question

Find the angle between the vectors $\vec a=\begin{pmatrix}2\\0\\4\end{pmatrix}$ and $\vec b=\begin{pmatrix}-3\\1\\2\end{pmatrix}$ , giving the angle in degrees correct to 1 decimal place.

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Solution

Criteria	Marks
Provides correct solution	3
Correctly finds the dot product and both magnitudes OR uses the correct dot product or a correct magnitude in $\mathbf a\cdot\mathbf b=	\mathbf a
Correctly finds the dot product of the two vectors OR correctly finds the magnitude of one vector	1

Sample answer:

\text{Dot product: }\begin{pmatrix}2\\0\\4\end{pmatrix}\cdot\begin{pmatrix}-3\\1\\2\end{pmatrix}&=-6+0+8\\&=2 \end{aligned}$$ Magnitudes: $$\left|\begin{pmatrix}2\\0\\4\end{pmatrix}\right|=\sqrt{2^2+0^2+4^2}=\sqrt{20}$$ $$\left|\begin{pmatrix}-3\\1\\2\end{pmatrix}\right|=\sqrt{(-3)^2+1^2+2^2}=\sqrt{14}$$ $$\sqrt{20}\sqrt{14}\cos\theta=2$$ $$\cos\theta=0.1195\dots$$ $$\theta=83.135^\circ\dots\approx83.1^\circ\text{ (1 d.p.)}$$

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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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