Exam 11: Vectors in the Plane

Suppose a plane flies at a constant groundspeed of 500 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction Must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to Maintain its groundspeed and eastward direction. Round your answer to two decimal places.

Find an equation in spherical coordinates for the equation given in rectangular coordinates. $y = 4$

Find the triple scalar product to find the volume of the parallelepiped having adjacent edges given by the vectors $\mathbf { u } = \{ - 4,9,5 \} , \mathbf { v } = \{ 3,5 , - 1 \} , \quad \mathbf { w } = \{ 8 , - 8,7 )$

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $r = 4 z$

Graph the solid described in spherical coordinates. $0 \leq p \leq 1,0 \leq \theta \leq \frac{\pi}{2}, \frac{\pi}{4} \leq \phi \leq \frac{\pi}{2}$

Identify the following quadric surface. $z = \frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 16 }$

Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places. $\mathbf { u } =(-7,9,5), \mathbf { v } = ( - 2 , - 2,5 )$

$\text { Suppose a plane is flying in the direction } 302 ^ { \circ } \text {. Its speed with respect to the air is } 900$ kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places.

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve $z = \frac { 1 } { 4 } y ^ { 2 } + 1 , ( 0 \leq y \leq 4 ) \text { in the } y z - \text { plane about }$ the z-axis. Find an equation for the surface of revolution.

$\text { Find the cross product of the unit vector } \mathbf { k } \times \mathbf { i } \text {. }$

Find the set of parametric equations of the line through the point $( - 3,6,3 ) \text { that is }$ parallel to the line $x = - 8 + 5 t , y = - 3 + 8 t$ , and $z = 4 + 4 t$ .

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point $( x , y , z )$ where $x = 7 + t , y = 3 - t , z = 1 + t$ . Also, at time $t$ , the second insect is at the point $( x , y , z )$ where $x = 4 + t , y = 6 + t , z = t$ . Assume distances are given in inches. Find the distance between the two insects at time $t = 0$ . Round your answer to two decimal places.

Find a set of parametric equations of the line that passes through the point $( - 2,5 , - 1 )$ and is parallel to the $x y$ -plane and $y z$ -plane.

Find an equation in cylindrical coordinates for the equation given in rectangular coordinates. $z = 9 x ^ { 2 } + 9 y ^ { 2 } - 2$

Determine whether the lines given below intersect, and, if so, find the point of intersection. x=-6+2t,y=9+4t,z=-13+3t x=4+2s,y=25+2s,z=2+3s

Find the distance between the point $( 1,1,1 ) \text { and the line given by the set of }$ parametric equations $x = 1 - 3 t , y = 3 + t , z = - t$ .

Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither. $\mathbf { u } = \{ 1 , - 3 \} , \mathbf { v } = \{ 2 , - 9 \}$

Find the angle between the vectors for u and v. $\mathbf { u } = 6 \mathbf { i } + 4 \mathbf { j } , \mathbf { v } = 5 \mathbf { i } + 3 \mathbf { j }$

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph.

Suppose a car is towed using a force of 1300 newtons. The chain used to pull the car makes a $25 ^ { \circ }$ angle with the horizontal. Find the work done in towing the car 4 kilometers. Round your answer to two decimal places.