In an assembly operation illustrated in Figure P3.30, a robot moves an object first straight upward and then also to the east, around an arc forming one-quarter of a circle of radius 4.80 cm that lies in an east–west vertical plane. The robot then moves the object upward and to the north, through one-quarter of a circle of radius 3.70 cm that lies in a north–south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical.

A minivan travels straight north in the right lane of a divided highway at 28.0 m/s. A camper passes the minivan and then changes from the left lane into the right lane. As it does so, the camper’s path on the road is a straight displacement at 8.508 east of north. To avoid cutting off the minivan, the north–south distance between the camper’s back bumper and the minivan’s front bumper should not decrease. (a) Can the camper be driven to satisfy this requirement? (b) Explain your answer.

A person walks \(25.0^{\circ}\) north of east for 3.10 km. How far would she have to walk due north and due east to arrive at the same location?

Your dog is running around the grass in your backyard. He undergoes successive displacements 3.50 m south, 8.20 m northeast, and 15.0 m west. What is the resultant displacement?

Given the vectors \(\overrightarrow{\mathbf{A}}\) = \(2.00 \hat{\mathbf{i}}+6.00 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{B}}\) = \(3.00 \hat{\mathbf{i}}-2.00 \hat{\mathbf{j}}\), (a) draw the vector sum \(\overrightarrow{\mathbf{C}}\) = \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) and the vector difference \(\overrightarrow{\mathbf{D}}\) = \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\). (b) Calculate \(\overrightarrow{\mathbf{C}}\) and \(\overrightarrow{\mathbf{D}}\), in terms of unit vectors. (c) Calculate \(\overrightarrow{\mathbf{C}}\) and \(\overrightarrow{\mathbf{D}}\) in terms of polar coordinates, with angles measured with respect to the positive x axis.

The helicopter view in Fig. P3.15 shows two people pulling on a stubborn mule. The person on the right pulls with a force \(\overrightarrow{\mathbf{F}}_1\) of magnitude 120 N and direction of \(\theta_1\) = \(60.0^{\circ}\). The person on the left pulls with a force \(\overrightarrow{\mathbf{F}}_2\) of magnitude 80.0 N and direction of \(\theta_2\) = \(75.0^{\circ}\). Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are Figure P3.15 measured in units of newtons (symbolized N).

A snow-covered ski slope makes an angle of \(35.0^{\circ}\) with the horizontal. When a ski jumper plummets onto the hill, a parcel of splashed snow is thrown up to a maximum displacement of 1.50 m at \(16.0^{\circ}\) from the vertical in the uphill direction as shown in Figure P3.16. Find the components of its maximum displacement (a) parallel to the surface and (b) perpendicular to the surface.

Consider the three displacement vectors \(\overrightarrow{\mathbf{A}}\) = \((3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}) \mathrm{m}\), \(\overrightarrow{\mathbf{B}}\) = \((\hat{\mathbf{i}}-4 \hat{\mathbf{j}}) \mathrm{m}\), and \(\overrightarrow{\mathbf{C}}\) = \((-2 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \mathrm{m}\). Use the component method to determine (a) the magnitude and direction of \(\overrightarrow{\mathbf{D}}\) = \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}\) and (b) the magnitude and direction of \(\overrightarrow{\mathbf{E}}\) = \(-\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}\).

Vector \(\overrightarrow{\mathbf{A}}\) has x and y components of -8.70 cm and 15.0 cm, respectively; vector \(\overrightarrow{\mathbf{B}}\) has x and y components of 13.2 cm and -6.60 cm, respectively. If \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+3 \overrightarrow{\mathbf{C}}\) = 0, what are the components of \(\overrightarrow{\mathbf{C}}\)?

The vector \(\overrightarrow{\mathbf{A}}\) has x, y, and z components of 8.00, 12.0, and -4.00 units, respectively. (a) Write a vector expression for \(\overrightarrow{\mathbf{A}}\) in unit-vector notation. (b) Obtain a unit-vector expression for a vector \(\overrightarrow{\mathbf{B}}\) one-fourth the length of \(\overrightarrow{\mathbf{A}}\) pointing in the same direction as \(\overrightarrow{\mathbf{A}}\). (c) Obtain a unit-vector expression for a vector \(\overrightarrow{\mathbf{C}}\) three times the length of \(\overrightarrow{\mathbf{A}}\) pointing in the direction opposite the direction of \(\overrightarrow{\mathbf{A}}\).

Given the displacement vectors \(\overrightarrow{\mathbf{A}}\) = \((3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{m}\) and \(\overrightarrow{\mathbf{B}}\) = \((2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}) \mathrm{m}\), find the magnitudes of the following vectors and express each in terms of its rectangular components. (a) \(\overrightarrow{\mathbf{C}}\) = \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) (b) \(\overrightarrow{\mathbf{D}}\) = \(2 \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\)

Vector \(\overrightarrow{\mathbf{A}}\) has a negative x component 3.00 units in length and a positive y component 2.00 units in length. (a) Determine an expression for \(\overrightarrow{\mathbf{A}}\) in unit-vector notation. (b) Determine the magnitude and direction of \(\overrightarrow{\mathbf{A}}\). (c) What vector \(\overrightarrow{\mathbf{B}}\) when added to \(\overrightarrow{\mathbf{A}}\) gives a resultant vector with no x component and a negative y component 4.00 units in length?

Three displacement vectors of a croquet ball are shown in Figure P3.22, where \(|\overrightarrow{\mathbf{A}}|\) = 20.0 units, \(|\overrightarrow{\mathbf{B}}|\) = 40.0 units, and \(|\overrightarrow{\mathbf{C}}|\) = 30.0 units. Find (a) the resultant in unit-vector nota- tion and (b) the magnitude and direc- tion of the resultant displacement.

(a) Taking \(\overrightarrow{\mathbf{A}}\) = \((6.00 \hat{\mathbf{i}}-8.00 \hat{\mathbf{j}})\) units, \(\overrightarrow{\mathbf{B}}\) = \((-8.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}})\) units, and \(\overrightarrow{\mathbf{C}}\) = \((26.0 \hat{\mathbf{i}}+19.0 \hat{\mathbf{j}})\) units, determine a and b such that \(a\overrightarrow{\mathbf{A}}+b\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}\) = 0. (b) A student has learned that a single equation cannot be solved to determine values for more than one unknown in it. How would you explain to him that both a and b can be determined from the single equation used in part (a)?

Vector \(\overrightarrow{\mathbf{B}}\) has x, y, and z components of 4.00, 6.00, and 3.00 units, respectively. Calculate (a) the magnitude of \(\overrightarrow{\mathbf{B}}\) and (b) the angle that \(\overrightarrow{\mathbf{B}}\) makes with each coordinate axis.

Use the component method to add the vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) shown in Figure P3.9. Both vectors have magnitudes of 3.00 m and vector \(\overrightarrow{\mathbf{A}}\) makes an angle of \(\theta\) = \(30.0^{\circ}\) with the x axis. Express the resultant \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) in unit-vector notation.

A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (a) What is her resultant displacement? (b) What is the total distance she travels?

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of \(120^{\circ}\) with the positive x axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of \(35.0^{\circ}\) to the positive x axis. Find the magnitude and direction of the second displacement.

Figure P3.28 illustrates typical proportions of male (m) and female (f) anatomies. The displacements \(\overrightarrow{\mathbf{d}}_{1 \mathrm{~m}}\) and \(\overrightarrow{\mathbf{d}}_{1 \mathrm{f}}\) from the soles of the feet to the navel have magnitudes of 104 cm and 84.0 cm, respectively. The displacements \(\overrightarrow{\mathbf{d}}_{2 \mathrm{~f}}\) and \(\overrightarrow{\mathbf{d}}_{2 \mathrm{~m}}\) from the navel to outstretched fingertips have magnitudes of 100 cm and 86.0 cm, respectively. Find the vector sum of these displacements \(\overrightarrow{\mathbf{d}}_{3}\) = \(\overrightarrow{\mathbf{d}}_{1}\) + \(\overrightarrow{\mathbf{d}}_{2}\) for both people.

Review. As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction \(60.0^{\circ}\) north of west with a speed of 41.0 km/h. (a) What is the unit-vector expression for the velocity of the hurricane? It maintains this velocity for 3.00 h, at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant 25.0 km/h. This new velocity is maintained for 1.50 h. (b) What is the unit-vector expression for the new velocity of the hurricane? (c) What is the unit-vector expression for the displacement of the hurricane during the first 3.00 h? (d) What is the unit-vector expression for the displacement of the hurricane during the latter 1.50 h? (e) How far from Grand Bahama is the eye 4.50 h after it passes over the island?

In an assembly operation illustrated in Figure P3.30, a robot moves an object first straight upward and then also to the east, around an arc forming one-quarter of a circle of radius 4.80 cm that lies in an east–west vertical plane. The robot then moves the object upward and to the north, through one-quarter of a circle of radius 3.70 cm that lies in a north–south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical.

Review. You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the x axis and at a fixed height of \(7.60 \times 10^3\) m. At time t = 0, the airplane is directly above you so that the vector leading from you to it is \(\overrightarrow{\mathbf{P}}_0\) = \(7.60 \times 10^3 \hat{\mathbf{j}}\) m. At t = 30.0 s, the position vector leading from you to the airplane is \(\overrightarrow{\mathbf{P}}_{30}\) = \(\left(8.04 \times 10^3 \hat{\mathbf{i}}+7.60 \times 10^3 \hat{\mathbf{j}}\right)\) m as suggested in Figure P3.31. Determine the magnitude and orientation of the airplane's position vector at t = 45.0 s.

Why is the following situation impossible? A shopper pushing a cart through a market follows directions to the canned goods and moves through a displacement \(8.00 \hat{\mathbf{i}} \mathrm{m}\) down one aisle. He then makes a \(90.0^{\circ}\) turn and moves 3.00 m along the y axis. He then makes another \(90.0^{\circ}\) turn and moves 4.00 m along the x axis. Every shopper who follows these directions correctly ends up 5.00 m from the starting point.

In Figure P3.33, the line segment represents a path from the point with position vector \((5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}) \mathrm{m}\) to the point with location \((16 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \mathrm{m}\). Point ? is along this path, a fraction f of the way to the destination. (a) Find the position vector of point ? in terms of f. (b) Evaluate the expression from part (a) for f = 0. (c) Explain whether the result in part (b) is reasonable. (d) Evaluate the expression for f = 1. (e) Explain whether the result in part (d) is reasonable.