Along the curve C traced out by the end point of the position vector tk (t) = (- sin ti + cos tj+t) = [0,π/2] calculate the line integral √ F(r) - di dr of the vector field F(x, y, z) = −y+xj+yzk.

4. Determine the directional derivatives of +V3 sin" xy $(x, y) =tan at the point (1,1) in the direction of the vector 31-2j .

A charged particle begins at rest at the origin. Suddenly, a force causes the particleto accelerate according to the vector function a(t) = ⟨ sin(t) , 6t , 2cos(t)⟩Find functions for the velocity, speed and position of the particle at time t

The path r(t)=(5 sin t) i+(5 cos t) j describes motion on the circle x^2+y^2=25. Find the​ particle's velocity and acceleration vectors at t=π/3 and π/6​, and sketch them as vectors on the curve.

Find all points on the curve r(t) = ti + t^2 j + t^3k where its tangent line is parallel to the vector 2i + 8j + 24k.

Let u(t) = 21'i+ (-1)j-8k. Compute the derivative of the following function. (11 +7t) u(t) Select the corect choice below and fill in the answer box(es) to complete your choice. O A. The derivative is the scalar function O B. The derivative is the vector-valued function Di+ i+ k. Click to select and enter your answer(s) and then click Check Answer

Consider the given vector equation. r(t) = 2e'i + 3e-j (a) Find r'(t). r(t) = (b) Sketch the plane curve together with the position vector r(t) and the tangent vector r'(t) for the given value of t= 0. y y

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The paraboloid z = 3x2 + y2 and the parabolic cylinder y = 2x2 r(t) = (1,322 + 9^) * Need Help? Read It

Find a vector function that represents the curve of intersection of two surfaces: z = x? - y? and x' + y² =1

An airplane took off from point (3, 0, 0) at an initial velocity, v(0)= 3j with an acceleration vector of a(t) = (-3 cos t)i + (-3 sin t)j + 2k. Determine the vector function, r(t), to represent the path of the airplane as a function of t.