Solved: A flying drone is programmed to complete a series of movements in a horizontal... [IB Mathematics Applications & Interpretation (AI)]

Solved: A flying drone is programmed to complete a series of movements in a horizontal... [IB Mathematics Applications & Interpretation (AI)] | RevisionDojo

A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes. In each case, the drone moves to a new position represented by the following transformations: - a rotation anticlockwise of π/6 radians about O - a reflection in the line y = x/√3 - a rotation clockwise of π/3 radians about O. All the movements are performed in the listed order.

Write down the matrix for each of the three transformations: (i) rotation anticlockwise of π/6 radians about O (ii) reflection in the line y = x/√3 (iii) rotation clockwise of π/3 radians about O

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Question

HLPaper 2

a rotation anticlockwise of π/6 radians about O
a reflection in the line y = x/√3
a rotation clockwise of π/3 radians about O. All the movements are performed in the listed order.

Write down the matrix for each of the three transformations:

(i) rotation anticlockwise of π/6 radians about O

(ii) reflection in the line y = x/√3

(iii) rotation clockwise of π/3 radians about O

[3]

Verified

Solution

(i) rotation anticlockwise $\frac{\pi}{6}$ is $\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right)$ (A1)

(ii) reflection in $y = \frac{x}{\sqrt{3}}$ is $\left(\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2}\end{array}\right)$ (A1)

(iii) rotation clockwise $\frac{\pi}{3}$ is $\left(\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\end{array}\right)$ (A1)

Find a single matrix P that defines a transformation that represents the overall change in position by multiplying the three matrices in the correct order.

[2]

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Solution

an attempt to multiply three matrices

$\boldsymbol{P}=\left(\begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{array}\right)\left(\begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array}\right)\left(\begin{array}{cc} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{array}\right)$ (M1) $\boldsymbol{P}=\left(\begin{array}{cc} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{array}\right) \quad \text{OR} \quad \left(\begin{array}{cc} 0.866 & -0.5 \\ -0.5 & -0.866 \end{array}\right)$ A1A1

Find P².

[1]

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Solution

$\boldsymbol{P}^{2}=\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2}\end{array}\right)\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2}\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ (M1)A1 Note: Do not award $\boldsymbol{A1}$ if final answer not resolved into the identity matrix $I$ .

Hence state what the value of P² indicates for the possible movement of the drone.

[1]

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Solution

if the overall movement of the drone is repeated the drone would return to its original position R1

Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A', B' and C' respectively.

Show that the area of triangle ABC is equal to the area of triangle A'B'C'.

[3]

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Solution

Method 1:

The determinant of matrix P is 1 (M1)
This means the transformation preserves area (A1)
Therefore triangle ABC has the same area as triangle A'B'C' (R1)

Method 2:

Area is preserved under rotations and reflections (M1)
Each individual transformation preserves area (A1)
Therefore the composite transformation preserves area (R1)

Find a single transformation that is equivalent to the three transformations represented by matrix P.

[1]

Verified

Solution

The matrix P represents a rotation of π radians (or 180°) about O (A1)

Note: Accept any clear indication that it's a half turn/180° rotation about the origin.

Mathematics Applications & Interpretation (AI) International Baccalaureate (IB) Practice Question: A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set o...