Exam 8: Polar Coordinates; Vectors

Solve the problem. -A plane is headed due south with an airspeed of 210 miles per hour. A wind from a direction of S30°W is blowing at 20 miles per hour. Find the groundspeed and resulting direction of the plane, rounded to the nearest Whole number.

Plot the point given in polar coordinates. - $\left(2,-\frac{\pi}{4}\right)$ A) B) C) D)

Solve the problem. -If $\mathrm { P } = ( - 6 , - 9 )$ and $\mathrm { Q } = ( \mathrm { x } , - 105 )$ , find all numbers $x$ such that the vector represented by $\overrightarrow { \mathrm { PQ } }$ has length $- 120$

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. -P = (4, 4); Q = (-3, -6)

Find the distance from $\mathrm { P } _ { 1 } \text { to } \mathrm { P } _ { 2 }$ - $\mathrm { P } _ { 1 } = ( - 3,0,4 ) \text { and } \mathrm { P } _ { 2 } = ( 2,3,1 )$

Find the indicated cross product. -v = 6i - 5j + 4k, w = -6i + 2j + 3k Find v × w.

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fourth roots of -16 A) $2 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) , 2 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) , 2 \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right) , 16 \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$ B) $\sqrt [ 4 ] { 2 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) , \sqrt [ 4 ] { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) , \sqrt [ 4 ] { 2 } \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right) , \sqrt [ 4 ] { 2 } \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$ C) $2 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 2 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right) , 2 \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right) , 2 \left( \cos 360 ^ { \circ } + i \sin 360 ^ { \circ } \right)$ D) $16 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) , 16 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) , 16 \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right) , 16 \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$

Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - $\mathrm { r } \sin \theta = 10$

Write the expression in the standard form a + bi. - $\left[ 2 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right) \right] ^ { 3 }$

The rectangular coordinates of a point are given. Find polar coordinates for the point. -(-2, 0.6) Round the polar coordinates to two decimal places, with θ in radians.

Match the point in polar coordinates with either A, B, C, or D on the graph. - $\left[-3, \frac{\pi}{3}\right]$

Solve the problem. -An audio speaker that weighs 50 pounds hangs from the ceiling of a restaurant from two cables as shown in the figure. To two decimal places, what is the tension in the two cables?

Describe the set of points (x, y, z) defined by the equation. -x = -2

Find the unit vector having the same direction as v. -v = -4j A) $\mathbf { u } = - 4 \mathbf { j }$ B) $\mathbf { u } = 16 \mathbf { j }$ C) $\mathbf { u } = - \frac { 1 } { 4 } \mathrm { j }$ D) $\mathbf { u } = - \mathbf { j }$

Perform the indicated operation. - =3+5-2 and =2-4+6 Find \|-\|\| A) B) -2 C) D) 3-6

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis. - $\| \mathrm { v } \| = 7 , \alpha = 270 ^ { \circ }$

Use the vectors in the figure below to graph the following vector. - $\mathbf{u}+\mathbf{z}$

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( - 5 , \frac { 3 \pi } { 4 } \right)$

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 5 , \frac { 3 \pi } { 4 } \right)$