Exam 21: Parameters, Coordinates, Integrals

Let $\vec { v } _ { 1 } = 2 \vec { i } - 5 \vec { j } + \vec { k }$ and $\vec { v } _ { 2 } = 5 \vec { i } + \vec { j } + \vec { k }$ Find a vector which is perpendicular to $\vec { v } _ { 1 }$ and $\vec { v } _ { 2 }$ to find an equation of the plane through the point (1, 2, -1)and with normal vector perpendicular to both $\vec { v } _ { 1 }$ and $\vec { v } _ { 2 }$ .Express your answer in the form $A x + B y + C z = D$

Consider the change of variables x = s + 3t, y = s - 2t. Let R be the region bounded by the lines 2x + 3y = 1, 2x + 3y = 4, x - y = -3, and x - y = 2.Find the region T in the st-plane that corresponds to region R. Use the change of variables to evaluate $\int _ { R } 2 x + 3 y d A$ .

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (5, 0, 4). Let P be the plane containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder). In each case, give a parameterization $( x ( \theta ) , y ( \theta ) , z ( \theta ) )$ and specify the range of values your parameters must take on. (i)the circle in which the cylinder, S, cuts the plane, P. (ii)the surface of the cylinder S.