The task is to answer the question below: Given that C₁ be the directed line segment from (2,0)to (−2, 0)and C₂ be the upper half of the circle x² + y² = 4 oriented clockwise. Let C = C₁ U C₂ and consider the vector field F~ (x, y) = (xy² + e²x − 3y,x²y — siny). (a) Evaluate S F.dR

Find all points on the curve r(t) = ti + t²j+t³k where its tangent line is parallel to the vector 21+ 8j + 24k.

Express the vector below in Cartesian coordinate: B = 2r sin cospar+r cose cose aer sind a ax= [2x/√(x² + y² + z²)] + {xz / [(x² + y²) * √(x². • √(x² + y² + z²)]} + {[y² * √(x² + y² + z²)] / (x² + y²)} ay = [xy² / √(x² + y² + z²)] + {[xyz²] / [(x² + y²) * √(x² + y² + z²)]} − {[xy * (x² + y² + z²)] / (x² + y²)} - ax= [2x² / √(x² + y² + z²)] + {xz / [(x² + y²) * √(x² + y² + z²)]} + {[y² * √(x² + y² + z²)] / (x² + y²)} a₂ = x²² / √(x² + y² + z²) az ay = [xy / √(x² + y² + z²)] + {[xyz²] / [(x² + y²) * √(x² + y² + z²)]} − {[xy * √(x² + y² + z²)] / (x² + y²)} a₂ = (xz)/√(x² + y² + z²) a₂ = (xz) / (x² + y² + z²) ax= [x² / √(x² + y² + z²)] + {xz / [(x² + y²) * √(x² + y² + z²)]} + {[y² * √(x² + y² + z²)] / (x + y)} ay = [2xy / √(x² + y² + z²)] + {[xyz²] / [(x² + y²) * √(x² + y² + z²)]} - {[xy * √(x² + y² + z²)] / (x² + y²)}

Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)

Consider the following function. T: R2 → R, T(x, y) = (2x2, 3xy, y?) Find the following images for vectors u = (u,, u2) and v = (v,, v2) in R2 and the scalar c. (Give all answers in terms of 1' "1, U2, V1, V2, and c.) T(u) T(v) T(u) + T(v) = T(u + v) CT(u) = T(cu) = Determine whether the function is a linear transformation. O linear transformation not a linear transformation

Suppose a car is traveling at 50 km/hr, and that the positive y-axis is north and the positive x-axis is east. Resolve the car's velocity vector (in 2-space) into components if the car is traveling in each of the following directions. The units are km/hr, but you do not need to enter units. 1. East: 2. South: 3. Southeast: 4. Northwest: