91Ó°ÊÓ

Find the component form of the sum of u and v with direction angles \(\boldsymbol{\theta}_{\mathrm{u}}\) and \(\boldsymbol{\theta}_{\mathrm{v}}\). $$\begin{array}{cc} \text{Magnitude} && \text{Angle} \\ \|\mathbf{u}\|=4 && \theta_{\mathrm{u}}=60^{\circ} \\ \|\mathbf{v}\|=4 && \theta_{\mathrm{v}}=90^{\circ} \end{array}$$

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

A tractor pulls a log 800 meters, and the tension in the cable connecting the tractor and log is approximately 15,691 newtons. The direction of the force is \(35^{\circ}\) above the horizontal. Approximate the work done in pulling the log.

Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of \(30^{\circ}, 45^{\circ},\) and \(120^{\circ},\) respectively, with the positive \(x\) -axis. Find the direction and magnitude of the resultant of these forces.

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.

Use vectors to prove that the diagonals of a rhombus are perpendicular.

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?

Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar.