Question 1

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?

Accepted Answer

The correct answer is A because the work done is equal to the force applied multiplied by the distance moved in the direction of the force. In this case, the force is 40 pounds, the distance moved is 30 feet, and the angle between the force and the distance is 22 degrees. So, the work done is 40 pounds * 30 feet * cos(22 degrees) = 1,112.6 ft-lb (rounded to the nearest tenth).

Question 2

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

Accepted Answer

The equation x = -2 defines a plane in three-dimensional space where the x-coordinate of every point on the plane is -2, and the y and z coordinates can be any real numbers. This means all points on this plane will have the form (-2, y, z).

Question 3

Solve the problem. Round your answer to the nearest tenth.
-Find the work done by a force of 200 pounds acting in the direction -i + 2j in moving an object 75 feet from (0, 0) to (-75, 0).

Accepted Answer

The answer of Solve the problem. Round your answer to...

Question 4

Solve the problem.
-An SUV weighing 4,400 pounds is parked on a street which has an incline of 15°. Find the force required to keep the SUV from rolling down the hill and the force of the SUV perpendicular to the hill. Round the forces to the
Nearest hundredth.

Accepted Answer

The force required to keep the SUV from rolling down the hill is $4400 \times \sin(15^\circ) \approx 1,138.80 \mathrm{lb}$, and the force perpendicular to the hill is $4400 \times \cos(15^\circ) \approx 4,250.07 \mathrm{lb}$.

Question 5

Find the distance from P1 to P2.
-$$\mathrm { P } _ { 1 } = ( 2,2 , - 1 ) \text { and } \mathrm { P } _ { 2 } = ( - 1 , - 3,2 )$$

Accepted Answer

The distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in 3D space is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$. Substituting the given points $P_1 = (2, 2, -1)$ and $P_2 = (-1, -3, 2)$ into the formula gives $\sqrt{(-1 - 2)^2 + (-3 - 2)^2 + (2 - (-1))^2} = \sqrt{(-3)^2 + (-5)^2 + 3^2} = \sqrt{9 + 25 + 9} = \sqrt{43}$.

Question 6

Solve the problem. Round your answer to the nearest tenth.
-Find the work done by a force of 6 pounds acting in the direction of 43° to the horizontal in moving an object 6 feet from (0, 0) to (6, 0).

Accepted Answer

The work done is calculated using the formula $W = Fd\cos(\theta)$, where $W$ is work, $F$ is the force, $d$ is the distance, and $\theta$ is the angle of the force with the direction of motion. Here, $F = 6$ pounds, $d = 6$ feet, and $\theta = 43^\circ$. Thus, $W = 6 \times 6 \times \cos(43^\circ) \approx 26.3$ ft-lb.

Question 7

Describe the set of points (x, y, z) defined by the equation.
-y = 0

Accepted Answer

The equation y = 0 defines a plane in three-dimensional space where the y-coordinate of every point on the plane is 0. This means that the x and z coordinates can be any value, hence all points of the form (x, 0, z) are included.

Question 8

Find the position vector for the vector having initial point P and terminal point Q.
-$$\mathrm { P } = ( - 1 , - 1,0 ) \text { and } \mathrm { Q } = ( - 3,3 , - 4 )$$

Accepted Answer

The position vector from P to Q is found by subtracting the coordinates of P from Q: $(-3 - (-1), 3 - (-1), -4 - 0) = (-2, 4, -4)$, which corresponds to $-2i + 4j - 4k$.

Question 9

Describe the set of points (x, y, z) defined by the equation.
-z = 0

Accepted Answer

The equation z = 0 defines a plane in three-dimensional space where the z-coordinate of every point on the plane is 0. This means that any value of x and y is allowed as long as z is 0, which is correctly described by option C, all points of the form (x, y, 0).

Question 10

Solve the problem. Round your answer to the nearest tenth.
-A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Accepted Answer

The answer of Solve the problem. Round your answer to...

Question 11

Find the position vector for the vector having initial point P and terminal point Q.
-$\mathrm { P } = ( - 3,2 , - 3 )$ and $\mathrm { Q } = ( 2,0,1 )$

Accepted Answer

The position vector from P to Q is found by subtracting the coordinates of P from Q: $ \mathbf { v } = (2 - ( - 3), 0 - 2, 1 - ( - 3)) = (5, -2, 4)$, which corresponds to $5 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }$.

Question 12

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

Accepted Answer

The equation defines a line in three-dimensional space where x is always -2 and z is always -4, allowing y to be any value. This corresponds to all points of the form (-2, y, -4).

Question 13

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$$\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }$$

Accepted Answer

To decompose $\mathbf{v}$ into $\mathbf{v}_1$ (parallel to $\mathbf{w}$) and $\mathbf{v}_2$ (orthogonal to $\mathbf{w}$), we use the projection formula for $\mathbf{v}_1 = \text{proj}_{\mathbf{w}}\mathbf{v}$ and then find $\mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1$. The correct components for $\mathbf{v}_1$ and $\mathbf{v}_2$ match the decomposition criteria, and option D provides the accurate calculation based on these principles.

Question 14

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$$\mathbf { v } = - 2 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { w } = 3 \mathbf { i } + \mathbf { j }$$

Accepted Answer

The answer of Decompose v into two vectors v1 and...

Question 15

Find the distance from P1 to P2.
-$$P _ { 1 } = ( 0 , - 4,4 ) \text { and } P _ { 2 } = ( - 1,1 , - 3 )$$

Accepted Answer

The answer of Find the distance from P1 to P2.
-\[P...

Question 16

Find the position vector for the vector having initial point P and terminal point Q.
-$\mathrm { P } = ( 0,0,0 )$ and $\mathrm { Q } = ( - 3,3,4 )$

Accepted Answer

The position vector for a vector from point P to point Q is found by subtracting the coordinates of P from Q. Since P = (0,0,0) and Q = (-3,3,4), the position vector is $\mathbf{v} = (-3-0)\mathbf{i} + (3-0)\mathbf{j} + (4-0)\mathbf{k} = -3\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$.

Question 17

Find the distance from P1 to P2.
-$$P _ { 1 } = ( - 3,1,1 ) \text { and } P _ { 2 } = ( 1,0 , - 3 )$$

Accepted Answer

The distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in 3D space is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$. Substituting the given points $P_1 = (-3, 1, 1)$ and $P_2 = (1, 0, -3)$ into the formula gives $\sqrt{(1 - (-3))^2 + (0 - 1)^2 + (-3 - 1)^2} = \sqrt{4^2 + (-1)^2 + (-4)^2} = \sqrt{16 + 1 + 16} = \sqrt{33}$.

Question 18

Find the distance from P1 to P2.
-$\mathrm { P } _ { 1 } = ( - 3,0,4 )$ and $\mathrm { P } _ { 2 } = ( 2,3,1 )$

Accepted Answer

The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in 3D space is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$. Applying this formula to the given points $P_1 = (-3,0,4)$ and $P_2 = (2,3,1)$, we get $\sqrt{(2 - (-3))^2 + (3 - 0)^2 + (1 - 4)^2} = \sqrt{5^2 + 3^2 + (-3)^2} = \sqrt{25 + 9 + 9} = \sqrt{43}$.

Question 19

Find the position vector for the vector having initial point P and terminal point Q.
-$$\mathrm { P } = ( 4,2,2 ) \text { and } \mathrm { Q } = ( - 4 , - 1 , - 2 )$$

Accepted Answer

The position vector from P to Q is found by subtracting the coordinates of P from Q: $(-4 - 4)i + (-1 - 2)j + (-2 - 2)k = -8i - 3j - 4k$.

Question 20

Find the distance from P1 to P2.
-$\mathrm { P } _ { 1 } = ( 0,0,0 )$ and $\mathrm { P } _ { 2 } = ( 2,4,3 )$

Accepted Answer

The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in 3D space is given by the formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. Substituting the given points, the distance is $\sqrt{(2-0)^2 + (4-0)^2 + (3-0)^2} = \sqrt{4 + 16 + 9} = \sqrt{29}$.

Quiz 8: Polar Coordinates; Vectors

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?

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

Solve the problem. Round your answer to the nearest tenth. -Find the work done by a force of 200 pounds acting in the direction -i + 2j in moving an object 75 feet from (0, 0) to (-75, 0) .

Solve the problem. -An SUV weighing 4,400 pounds is parked on a street which has an incline of 15°. Find the force required to keep the SUV from rolling down the hill and the force of the SUV perpendicular to the hill. Round the forces to the Nearest hundredth.

Find the distance from P1 to P2. - $\mathrm { P } _ { 1 } = ( 2,2 , - 1 ) \text { and } \mathrm { P } _ { 2 } = ( - 1 , - 3,2 )$

Solve the problem. Round your answer to the nearest tenth. -Find the work done by a force of 6 pounds acting in the direction of 43° to the horizontal in moving an object 6 feet from (0, 0) to (6, 0) .

Describe the set of points (x, y, z) defined by the equation. -y = 0

Find the position vector for the vector having initial point P and terminal point Q. - $\mathrm { P } = ( - 1 , - 1,0 ) \text { and } \mathrm { Q } = ( - 3,3 , - 4 )$

Describe the set of points (x, y, z) defined by the equation. -z = 0

Solve the problem. Round your answer to the nearest tenth. -A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Find the position vector for the vector having initial point P and terminal point Q. - $\mathrm { P } = ( - 3,2 , - 3 )$ and $\mathrm { Q } = ( 2,0,1 )$

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

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }$

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = - 2 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { w } = 3 \mathbf { i } + \mathbf { j }$

Find the distance from P1 to P2. - $P _ { 1 } = ( 0 , - 4,4 ) \text { and } P _ { 2 } = ( - 1,1 , - 3 )$

Find the position vector for the vector having initial point P and terminal point Q. - $\mathrm { P } = ( 0,0,0 )$ and $\mathrm { Q } = ( - 3,3,4 )$

Find the distance from P1 to P2. - $P _ { 1 } = ( - 3,1,1 ) \text { and } P _ { 2 } = ( 1,0 , - 3 )$

Find the distance from P1 to P2. - $\mathrm { P } _ { 1 } = ( - 3,0,4 )$ and $\mathrm { P } _ { 2 } = ( 2,3,1 )$

Find the position vector for the vector having initial point P and terminal point Q. - $\mathrm { P } = ( 4,2,2 ) \text { and } \mathrm { Q } = ( - 4 , - 1 , - 2 )$

Find the distance from P1 to P2. - $\mathrm { P } _ { 1 } = ( 0,0,0 )$ and $\mathrm { P } _ { 2 } = ( 2,4,3 )$

Quiz 8: Polar Coordinates; Vectors

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?

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

Solve the problem. Round your answer to the nearest tenth. -Find the work done by a force of 200 pounds acting in the direction -i + 2j in moving an object 75 feet from (0, 0) to (-75, 0) .

Solve the problem. -An SUV weighing 4,400 pounds is parked on a street which has an incline of 15°. Find the force required to keep the SUV from rolling down the hill and the force of the SUV perpendicular to the hill. Round the forces to the Nearest hundredth.

Find the distance from P1 to P2. - P1=(2,2,−1) and P2=(−1,−3,2) \mathrm { P } _ { 1 } = ( 2,2 , - 1 ) \text { and } \mathrm { P } _ { 2 } = ( - 1 , - 3,2 ) P1​=(2,2,−1) and P2​=(−1,−3,2)

Solve the problem. Round your answer to the nearest tenth. -Find the work done by a force of 6 pounds acting in the direction of 43° to the horizontal in moving an object 6 feet from (0, 0) to (6, 0) .

Describe the set of points (x, y, z) defined by the equation. -y = 0

Find the position vector for the vector having initial point P and terminal point Q. - P=(−1,−1,0) and Q=(−3,3,−4) \mathrm { P } = ( - 1 , - 1,0 ) \text { and } \mathrm { Q } = ( - 3,3 , - 4 ) P=(−1,−1,0) and Q=(−3,3,−4)

Describe the set of points (x, y, z) defined by the equation. -z = 0

Solve the problem. Round your answer to the nearest tenth. -A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Find the position vector for the vector having initial point P and terminal point Q. - P=(−3,2,−3) \mathrm { P } = ( - 3,2 , - 3 ) P=(−3,2,−3) and Q=(2,0,1) \mathrm { Q } = ( 2,0,1 ) Q=(2,0,1)

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

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - v=−3i−5j,w=2i−j\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }v=−3i−5j,w=2i−j

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - v=−2i−2j,w=3i+j\mathbf { v } = - 2 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { w } = 3 \mathbf { i } + \mathbf { j }v=−2i−2j,w=3i+j

Find the distance from P1 to P2. - P1=(0,−4,4) and P2=(−1,1,−3) P _ { 1 } = ( 0 , - 4,4 ) \text { and } P _ { 2 } = ( - 1,1 , - 3 ) P1​=(0,−4,4) and P2​=(−1,1,−3)

Find the position vector for the vector having initial point P and terminal point Q. - P=(0,0,0) \mathrm { P } = ( 0,0,0 ) P=(0,0,0) and Q=(−3,3,4) \mathrm { Q } = ( - 3,3,4 ) Q=(−3,3,4)

Find the distance from P1 to P2. - P1=(−3,1,1) and P2=(1,0,−3) P _ { 1 } = ( - 3,1,1 ) \text { and } P _ { 2 } = ( 1,0 , - 3 ) P1​=(−3,1,1) and P2​=(1,0,−3)

Find the distance from P1 to P2. - P1=(−3,0,4) \mathrm { P } _ { 1 } = ( - 3,0,4 ) P1​=(−3,0,4) and P2=(2,3,1) \mathrm { P } _ { 2 } = ( 2,3,1 ) P2​=(2,3,1)

Find the position vector for the vector having initial point P and terminal point Q. - P=(4,2,2) and Q=(−4,−1,−2) \mathrm { P } = ( 4,2,2 ) \text { and } \mathrm { Q } = ( - 4 , - 1 , - 2 ) P=(4,2,2) and Q=(−4,−1,−2)

Find the distance from P1 to P2. - P1=(0,0,0) \mathrm { P } _ { 1 } = ( 0,0,0 ) P1​=(0,0,0) and P2=(2,4,3) \mathrm { P } _ { 2 } = ( 2,4,3 ) P2​=(2,4,3)