Which diagram best shows the curve described by the position vector
$r(t) = -5 ext{cos}(t) extbf{i} + 5 ext{sin}(t) extbf{j} + k$ for $0 ext{ } \leq t \leq 4\pi$. - HSC - SSCE Mathematics Extension 2 - Question 7 - 2021 - Paper 1
Question 7
Which diagram best shows the curve described by the position vector
$r(t) = -5 ext{cos}(t) extbf{i} + 5 ext{sin}(t) extbf{j} + k$ for $0 ext{ } \leq t \leq 4\pi$.
Worked Solution & Example Answer:Which diagram best shows the curve described by the position vector
$r(t) = -5 ext{cos}(t) extbf{i} + 5 ext{sin}(t) extbf{j} + k$ for $0 ext{ } \leq t \leq 4\pi$. - HSC - SSCE Mathematics Extension 2 - Question 7 - 2021 - Paper 1
Step 1
Analyze the Position Vector
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The position vector can be broken down as follows:
The x-component is given by x=−5cos(t).
The y-component is given by y=5sin(t).
The z-component is z=t.
The terms in the x and y components indicate that the motion is circular, due to the cosine and sine functions.
Step 2
Determine the Range of Motion
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
For 0≤t≤4π, we will complete two full revolutions around the origin (0, 0) in the x-y plane since the fundamental period of both cos(t) and sin(t) is 2π.
Step 3
Visualize the 3D Motion
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The z-component increases linearly with t, implying that as the point makes circular movements in the x-y plane, it also moves upward along the z-axis. Therefore, the complete motion describes a helix.
Step 4
Select the Correct Diagram
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
From the options provided, diagram D is the one that best represents a helix, showing the circular motion in the x-y plane along with the upward movement in the z-axis.
