Question

A particle moves along a curve whose parametric equations are x = e^-t, y = 2 cos t, z = sin t. Its velocity is:
1. \( - {e^{ - t}}\hat i - 2\sin t\,\hat j + \cos t\,\hat k\)
2. \( - {e^{ - t}}\hat i + 2\cos t\,\hat j + \sin t\,\hat k\)
3. \( {e^t}\hat i - 2\sin t\,\hat j + \cos t\,\hat k\)
4. \( {e^t}\hat i + - 2\sin t\,\hat j - \cos t\,\hat k\)

Answer 1

Correct Answer - Option 1 : \( - {e^{ - t}}\hat i - 2\sin t\,\hat j + \cos t\,\hat k\)

Concept:

when a particle moves along a curve whose parameter equation is given then velocity will be

(1) \(V = v_x \hat i + v_y \hat j + v_z \hat k\)

where v_x, v_y and v_z are velocities along x-axis, y-axis, and z-axis respectively,

Given, 

\(x = e^{-t} \; then\; v_x = \frac{dx}{dt} = \frac{d}{dt}(e^{-t}) = - e^{-t}\)

\(y = 2 cos t, \; v_y = \frac{d}{dt}(2cos x)=-2 sin t\)

\(z = sin t, \; v_z = \frac {d}{dt}(sint)= cos t\)

then, velocity

\(V = - {e^{ - t}}\hat i - 2\sin t\,\hat j + \cos t\,\hat k\)