91Ó°ÊÓ

Find a value of \(b\) so that \(15 \mathbf{i}-3 \mathbf{j}\) and \(-4 \mathbf{i}+b \mathbf{j}\) are orthogonal.

Prove that the projection of \(\mathbf{v}\) onto \(\mathbf{i}\) is \((\mathbf{v} \cdot \mathbf{i}) \mathbf{i}\).

Solve each equation in the complex number system. Express solutions in polar and rectangular form. $$x^{5}-32 i=0$$

Solve each equation in the complex number system. Express solutions in polar and rectangular form. $$x^{3}-(1+i \sqrt{3})=0$$

The wind is blowing at 10 knots. Sailboat racers look for a sailing angle to the 10 -knot wind that produces maximum sailing speed. In this application, \((r, \theta)\) describes the sailing speed, \(r,\) in knots, at an angle \(\theta\) to the 10 -knot wind. Use this information to solve Exercises. Interpret the polar coordinates: \(\left(6.3,50^{\circ}\right)\).

Find two vectors \(\mathbf{v}\) and \(\mathbf{w}\) such that the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is \(\mathbf{v}\).

Use a graphing utility to graph \(r=1+2 \sin n \theta\) for \(n=1,2,3,4,5,\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of large and small petals that occur corresponding to each value of \(n\) ? How are the large and small petals related when \(n\) is odd and when \(n\) is even?

A plane is flying at a speed of 320 miles per hour on a bearing of \(\mathrm{N} 70^{\circ} \mathrm{E} .\) Its ground speed is 370 miles per hour and its true course is \(30^{\circ} .\) Find the speed, to the nearest mile per hour, and the direction angle, to the nearest tenth of a degree, of the wind.

Solve each equation in the complex number system. Express solutions in polar and rectangular form. $$x^{3}-(1-i \sqrt{3})=0$$

Group members should research and present a report on unusual and interesting applications of vectors.