Exam 8: Polar Coordinates; Vectors

Solve the problem. -Two forces, $\mathbf { F } _ { 1 }$ of magnitude 35 newtons (N) and $\mathrm { F } _ { 2 }$ of magnitude 55 newtons, act on an object at angles of $45 ^ { \circ }$ and $- 60 ^ { \circ }$ (respectively) with the positive $x$ -axis. Find the direction and magnitude of the resultant force; that is, find $\mathbf { F } _ { \mathbf { 1 } } + \mathbf { F } _ { \mathbf { 2 } }$ . Round the direction and magnitude to two decimal places.

Solve the problem. Leave your answer in polar form. - z=5 20+i20 w=4 5+i5 Find $\frac { z } { w }$ .

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

Use the figure below. Determine whether the given statement is true or false. -A + H = F

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $y = x$

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

Describe the set of points (x, y, z) defined by the equation. -x = -2 and z = -4

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 6 + 2 \cos \theta ; \text { the pole }$

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 7 , \frac { 2 \pi } { 3 } \right)$

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

Solve the problem. -A box of supplies that weighs 1500 kilograms is suspended by two cables as shown in the figure. To two decimal places, what is the tension in the two cables?

Solve the problem. -Which of the following vectors is parallel to v = -10i - 8j?

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 = (2, -6); Q = (-3, 6)

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $x = - 3$

Solve the problem. -Plot the point $\left( 4 , \frac { \pi } { 6 } \right)$ and find other polar coordinates $( r , \theta )$ of the point for which: (a) $\mathrm { r } > 0 , \quad - 2 \pi \leq \theta < 0$ (b) $\mathrm { r } < 0 , \quad 0 \leq \theta < 2 \pi$ (c) $\mathrm { r } > 0 \quad 2 \pi \leq \theta < 4 \pi$

Solve the problem. Round your answer to the nearest tenth. -A person is pulling a freight cart with a force of 40 pounds. How much work is done in moving the cart 30 feet if the cart's handle makes an angle of 22° with the ground?

Solve the problem. -If $\mathbf { v } = 9 \mathbf { i } + 3 \mathbf { j }$ , what is $\| 8 \mathbf { v } \|$ ?

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

Solve the problem. Leave your answer in polar form. - z=5 3+i3 w=2 4+i4 Find zw.

Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = 3 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k }$