Which diagram best shows the curve described by the position vector
r(t) = -5cos(t)i + 5sin(t)j + k for 0 ≤ t ≤ 4π?
A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2021 - Paper 1
Question 7
Which diagram best shows the curve described by the position vector
r(t) = -5cos(t)i + 5sin(t)j + k for 0 ≤ t ≤ 4π?
A.
B.
C.
D.
Worked Solution & Example Answer:Which diagram best shows the curve described by the position vector
r(t) = -5cos(t)i + 5sin(t)j + k for 0 ≤ t ≤ 4π?
A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2021 - Paper 1
Step 1
Evaluate the position vector
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Answer
The position vector given is
extbfr(t)=−5extcos(t)extbfi+5extsin(t)extbfj+extbfk
This indicates a circular motion in the x-y plane, centered along the z-axis.
Step 2
Analyze the components
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Answer
The x-component, x=−5extcos(t), and the y-component, y=5extsin(t), suggest that as t varies from 0 to 4π, the path oscillates in the x-y plane while maintaining a constant z value (i.e., z = k).
Step 3
Determine the shape of the curve
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Answer
The trajectory forms a circle with radius 5 in the xy-plane, flipping direction due to the negative sign in the x-component. The z-component contributes a constant elevation.
Step 4
Select the correct diagram
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Answer
The correct answer is diagram D, which depicts a circular path in the x-y plane with a constant z elevation.
