Show — 3 marks
A drone operator is planning a delivery route. The drone starts at point A and flies 80 m due east, then 60 m due north to reach point B. The operator wants to know the direct displacement from A to B and the direction of travel.
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(a) Show that the magnitude of the displacement from A to B is 100 m.
[1 mark]
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(b) Calculate the angle that the direct displacement makes with the east direction. Give your answer to 1 decimal place.
[1 mark]
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(c) The drone must return to point A. Describe how the displacement vector for the return journey compares to the displacement vector from A to B.
[1 mark]
Show mark scheme
- (a) Uses Pythagoras' theorem: displacement = √(80² + 60²) = √(6400 + 3600) = √10000 = 100 m
- (b) Uses trigonometry: tan(θ) = 60/80 or sin(θ) = 60/100 or cos(θ) = 80/100 and calculates θ = 36.9° (or 37°)
- (c) The return displacement vector is equal in magnitude (100 m) but opposite in direction / is the negative of the A to B vector / has a bearing/angle of 216.9° (or equivalent description showing reversal of direction)
Compare — 2 marks
A drone pilot is planning two flight paths to deliver packages across a city. Path A requires the drone to fly 200 m due north, then 150 m due east. Path B requires the drone to fly 150 m due east, then 200 m due north. Both paths start from the same location.
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Compare the final displacement vectors for Path A and Path B.
[2 marks]
Show mark scheme
- Both paths result in the same final displacement (same magnitude and direction) / Both paths produce identical resultant vectors
- The order of vector addition does not affect the final result / This demonstrates the commutative property of vector addition
Calculate — 2 marks
Shape A is drawn on a coordinate grid. Shape A is translated by the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ to give Shape B.
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(a) One vertex of Shape A is at the point (1, 5). Calculate the coordinates of the corresponding vertex of Shape B.
[1 mark]
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(b) Another vertex of Shape B is at the point (7, 2). Calculate the coordinates of the corresponding vertex of Shape A.
[1 mark]
Show mark scheme
- (a) (4, 3) OR x = 4 and y = 3
- (b) (4, 4) OR x = 4 and y = 4
Show — 3 marks
Three points P, Q, and R are plotted on a coordinate grid. Point P has coordinates (4, 5), point Q has coordinates (7, 3), and point R has coordinates (10, 1).
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(a) Write down the vector $\overrightarrow{PQ}$ as a column vector.
[1 mark]
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(b) Show that P, Q, and R lie on a straight line.
[2 marks]
Show mark scheme
- (a) $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ oe
- (b) Work out vector $\overrightarrow{QR} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$
- (b) Show vectors are equal and conclude points are collinear
Explain — 2 marks
A triangle ABC is drawn on a coordinate grid. Point A has coordinates (2, 3), point B has coordinates (5, 3), and point C has coordinates (2, 6). The triangle is translated by the vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$ to form triangle A'B'C'.
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(a) Write down the coordinates of the point A'.
[1 mark]
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(b) Jenny says that the side A'B' will be longer than the side AB. Explain why Jenny is incorrect.
[1 mark]
Show mark scheme
- (a) (6, 1) (accept x = 6, y = 1)
- (b) Translation does not change the size of a shape / translation preserves lengths / A'B' = AB (accept equivalent statements)
GCSE Perfect