Find the principal unit normal vector N for the curve r(t).r(t) =(6t cos t – 6 sin t)j + (6t sin t + 6 cos t)k

For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the given parameter value r(t) = (7t2 – 7t)i + (t + 3)j + k ; t0 = 2

The position vector of a particle is r(t). Find the requested vector.The acceleration at t = 0 for r(t) = t2i + (4t3 – 9)j + k

A small sculpture made of brass is believed to have a secret central cavity. The weight of the sculpture in air is 15.76 N. When it is submerged in water, the weight is 13.86 N. What is the volume of the secret cavity?

Find the domain and range and describe the level curves for the function f(x,y).f(x, y) =

Solve the problem.A force of magnitude 14 pounds pulling on a suitcase makes an angle of 60° with the ground. Express the force in terms of its i and j components.

Match the equation with the surface it defines.x2 + y2 = 1

Solve the problem.Find the magnitude of the torque in foot-pounds at point P for the following lever: = 3 in. and = 10 lb

Solve the problem. Unless stated otherwise, assume that the projectile is launched from the origin over a horizontal surfaceAn ideal projectile is launched from level ground at a launch angle of 26° and an initial speed of 48 m/sec. How far away from the launch point does the projectile hit the ground?

Use implicit differentiation to find the specified derivative at the given point.Find at the point (1, -1) for 5xy2 – 4x2y + 2x = 0.