91Ó°ÊÓ

True or False?, determine whether the statement is true or false. Justify your answer. In addition to SSS and SAS, the Law of Cosines can be used to solve triangles with AAS conditions.

Decomposing a Vector into Components In Exercises \(59-62,\) find the projection of \(u\) onto \(v .\) Then write \(u\) as the sum of two orthogonal vectors, one of which is \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). $$\mathbf{u}=\langle- 3,-2\rangle$$ $$\mathbf{v}=\langle- 4,-1\rangle$$

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[5(\cos 3.2+i \sin 3.2)]^{4}$$

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.

Using the Law of Cosines In Exercises 79 and 80 , use the Law of Cosines to find the angle \(\alpha\) between the vectors. (Assume \(0^{\circ} \leq \alpha \leq 180^{\circ} . )\) $$\mathbf{v}=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=2 \mathbf{i}-\mathbf{j}$$

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.) \(\begin{array}{ll}{\text { Force } 1} & {\text { Force } 2} & {\text { Resultant Force }} \\ {45 \text { pounds }} & {60 \text { pounds }} & {90 \text { pounds }}\end{array}\)

Velocity A gun with a muzzle velocity of 1200 feet per second is fired at an angle of \(6^{\circ}\) above the horizontal. Find the vertical and horizontal components of the velocity.

Resultant Force Three forces with magnitudes of 70 pounds, 40 pounds, and 60 pounds act on an object at angles of \(-30^{\circ}, 45^{\circ}\) and \(135^{\circ}\) respectively, with the positive \(x\) -axis. Find the direction and magnitude of the resultant of these forces.

A commercial jet is flying from Miami to Seattle. The jet's velocity with respect to the air is 580 mile. 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?