91Ó°ÊÓ

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(\left[2\left(\cos \dfrac{\pi}{2} + i\ \sin \dfrac{\pi}{2}\right)\right]^8\)

In Exercises 75-78, find the component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) with direction angles \(\mathbf{\theta_u}\) and \(\mathbf{\theta_v}\). Magnitude ||\(\small{\mathbf{u}}\)|| \(= 4\) ||\(\small{\mathbf{v}}\)|| \(= 4\) Angle \(\mathbf{\theta_u} = 60^{\circ}\) \(\mathbf{\theta_v} = 90^{\circ}\)

BRAKING LOAD A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of \(10^{\circ}\). Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill.

WORK A force of 45 pounds exerted at an angle of \(30^{\circ}\) above the horizontal is required to slide a table across a floor (see figure). The table is dragged 20 feet. Determine the work done in sliding the table.

RESULTANT FORCE In Exercises 81 and 82, find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive \(x\)-axis and force 2 as a vector at an angle \(\theta\) with the positive \(x\)-axis.) Force 1 - 45 pounds Force 2 - 60 pounds Resultant Force - 90 pounds

NAVIGATION A commercial jet is flying from Miami to Seattle. The jet's velocity with respect to the air is 580 miles per hour, and its bearing is \(332^{\circ}\). The wind, at the altitude of the plane, is blowing from the southwest with a velocity of 60 miles per hour. (a) Draw a figure that gives a visual representation of the situation. (b) Write the velocity of the wind as a vector in component form. (c) Write the velocity of the jet relative to the air in component form. (d) What is the speed of the jet with respect to the ground? (e) What is the true direction of the jet?

CAPSTONE The initial and terminal points of vector are \((3, -4)\) and \((9, 1)\), respectively. (a) Write \(\mathbf{v}\) in component form. (b) Write \(\mathbf{v}\) as the linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). (c) Sketch \(\mathbf{v}\) with its initial point at the origin. (d) Find the magnitude of \(\mathbf{v}\).