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
Analyze the Position Vector
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Answer
The position vector is given by:
r(t)=−5extcos(t)i+5extsin(t)j+k
From this, we can observe that the components in the x and y directions depend on the cosine and sine functions, implying a circular motion when projected onto the xy-plane.
Step 2
Parameter Ranges
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Answer
The parameter t varies from 0 to 4π. This range corresponds to two complete cycles of the trigonometric functions, which will result in the point returning to its starting position twice.
Step 3
Identify the Curve Type
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Answer
As t goes from 0 to 4π, the circle described by the x and y components will be traced out twice. The z-component remains constant (k), meaning that it is a helical path that repeats vertically as it circles.
Step 4
Select the Correct Diagram
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Answer
Based on the helical description and confirming that the curve repeats the circular path specifically along the z-axis, the choice that best illustrates this behavior is Diagram D.
