If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
-Find the velocity vector. r(t) = (cot t)i + (csc t)j
A) $\mathbf { v } = \left( - \sec ^ { 2 } t \right) \mathbf { i } - ( \tan t \sec t ) \mathbf { j }$
B) $\mathbf { v } = \left( \csc ^ { 2 } \mathrm { t } \right) \mathbf { i } + ( \cot \mathrm { t } \csc \mathrm { t } ) \mathbf { j }$
C) $\mathbf { v } = \left( - \csc ^ { 2 } \mathrm { t } \right) \mathbf { i } - ( \cot \mathrm { t } \csc \mathrm { t } ) \mathbf { j }$
D) $\mathbf { v } = \left( \sec ^ { 2 } \mathrm { t } \right) \mathbf { i } + ( \tan \mathrm { t } \sec \mathrm { t } ) \mathbf { j }$

Question

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
-Find the velocity vector. r(t) = (cot t)i + (csc t)j 
A) $\mathbf { v } = \left( - \sec ^ { 2 } t \right) \mathbf { i } - ( \tan t \sec t ) \mathbf { j }$
B) $\mathbf { v } = \left( \csc ^ { 2 } \mathrm { t } \right) \mathbf { i } + ( \cot \mathrm { t } \csc \mathrm { t } ) \mathbf { j }$
C) $\mathbf { v } = \left( - \csc ^ { 2 } \mathrm { t } \right) \mathbf { i } - ( \cot \mathrm { t } \csc \mathrm { t } ) \mathbf { j }$
D) $\mathbf { v } = \left( \sec ^ { 2 } \mathrm { t } \right) \mathbf { i } + ( \tan \mathrm { t } \sec \mathrm { t } ) \mathbf { j }$

Quizplus · Accepted Answer

The velocity vector is the derivative of the position vector with respect to time.

Taking the derivative of r(t), we get:

r'(t) = (-csc^2t)i - (cot t csc t)j

Therefore, the velocity vector is:

v = r'(t) = (-csc^2t)i - (cot t csc t)j

which matches choice C.

If R(t) Is the Position Vector of a Particle in the Plane