Exam 17: Parameterization and Vector Fields

Consider the curve $\vec { r } ( t ) = \left( 5 t ^ { 2 } + 1 \right) \vec { i } + \left( t ^ { 2 } - 5 \right) \vec { j } + t \vec { k }$ Find the equation of the tangent line at the point where t = 2.

The equation $\vec { r } = 2 \vec { i } + 5 \vec { j } + 3 \vec { k } + t \vec { i } - \vec { j } + 2 \vec { k } )$ parameterizes a line through the point (4, 3, 7). What is the value of t at this point?

Find a parameterization of a curve that looks like sin y = z when viewed from the x-axis, and looks like x = z² when viewed from the y-axis.See the shadows drawn on the planes in the following picture. What does the curve look like when viewed from the z-axis?

Describe the similarities and differences between the following two curves. $\text { Curve 1: } \vec { r } ( t ) = ( 3 + 3 t ) \vec { i } + ( 1 - t ) \vec { j } + ( 3 + 4 t ) \vec { k } , - \infty \leq t \leq \infty \text {, }$ $\text { Curve 2: } \vec { r } ( t ) = \left( 3 + 3 t ^ { 2 } \right) \vec { i } + \left( 1 - t ^ { 2 } \right) \vec { j } + \left( 3 + 4 t ^ { 2 } \right) \vec { k } , - \infty \leq t \leq \infty \text {. }$

A particle moves at a constant speed along a line through P = (10,-20, 22)and Q = (22, -46, 46).Find a parametric equation for the line if the particle passes through P at time t = 3 and passes through Q at time t = 7.

For the following vector field, identify which one of the following formulas could represent it. The scales in the x and y directions are the same.No reasons need be given.