91Ó°ÊÓ

Consider vectors \(\mathbf{a}=\langle 2,-4\rangle, \mathbf{b}=\langle-1,2\rangle\), and \(\mathrm{c}=0\) Determine the scalars \(\alpha\) and \(\beta\) such that \(\mathbf{c}=\alpha \mathbf{a}+\beta \mathbf{b}\).

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of 70 mph and an angle of elevation of \(30^{\circ}\). Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

[I] Consider the torus of equation \(\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4 R^{2}\left(x^{2}+y^{2}\right)\), where \(R \geq r>0\). a. Write the equation of the torus in spherical coordinates. b. If \(R=r\), the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is \(\rho=2 R \sin \varphi\) c. Use a CAS to graph the horn torus with \(R=r=2\) in spherical coordinates.

For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. \(x-5 y-z=1,5 x-25 y-5 z=-3\)

Three forces act on object. Two of the forces have the magnitudes \(58 \mathrm{~N}\) and \(27 \mathrm{~N}\), and make angles \(53^{\circ}\) and \(152^{\circ}\), respectively, with the positive \(x\) -axis. Find the magnitude and the direction angle from the positive \(x\) -axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. Show that the lines of equations \(x=t, y=1+t, z=2+t, t \in \mathbb{R}\), and \(\frac{x}{2}=\frac{y-1}{3}=z-3\) are skew, and find the distance between them.

An airplane is flying in the direction of \(43^{\circ}\) east of north (also abbreviated as N43E) at a speed of 550 mph. A wind with speed 25 mph comes from the southwest at a bearing of N15E. What are the ground speed and new direction of the airplane?

A 50 -lb weight is hung by a cable so that the two portions of the cable make angles of \(40^{\circ}\) and \(53^{\circ}\), respectively, with the horizontal. Find the magnitudes of the forces of tension \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2}\) in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves \(3 \mathrm{ft}\) to the right, and falls \(5 \mathrm{ft}\) away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.

A guy-wire supports a pole that is \(75 \mathrm{ft}\) high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground \(50 \mathrm{ft}\) from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is \(50 \mathrm{lb}\). (Round to the nearest integer.)