Section 1
Introduction to Vectors
Find the resultant vectors if the given changes are made in the indicated examples of this section.In Example $2,$ what is the resultant of the three vectors if the direction of vector $\mathbf{A}$ is reversed?
Find the resultant vectors if the given changes are made in the indicated examples of this section.In Example $4,$ for vectors $\mathbf{A}$ and $\mathbf{B},$ what is vector $2 \mathbf{A}+3 \mathbf{B} ?$
Find the resultant vectors if the given changes are made in the indicated examples of this section.In Example $5,$ for vectors $\mathbf{A}$ and $\mathbf{B},$ what is vector $2 \mathbf{B}-\mathbf{A} ?$
Find the resultant vectors if the given changes are made in the indicated examples of this section.In Example $6,$ if $20^{\circ}$ replaces $40^{\circ},$ what is the resultant force?
Determine whether a scalar or a vector is described in (a) and (b). Explain your answers.(a) A soccer player runs $15 \mathrm{m}$ from the center of the field.(b) A soccer player runs 15 m from the center of the field toward the opponents' goal.
Determine whether a scalar or a vector is described in (a) and (b). Explain your answers.(a) A small-craft warning reports $25 \mathrm{mi} / \mathrm{h}$ winds.(b) A small-craft warning reports $25 \mathrm{mi} / \mathrm{h}$ winds from the north.
Determine whether a scalar or a vector is described in (a) and (b). Explain your answers.(a) An arm of an industrial robot pushes with a 10-lb force downward on a part.(b) A part is being pushed with a 10-lb force by an arm of an industrial robot.
Determine whether a scalar or a vector is described in (a) and (b). Explain your answers.(a) A ballistics test shows that a bullet hit a wall at a speed of $400 \mathrm{ft} / \mathrm{s}$(b) A ballistics test shows that a bullet hit a wall at a speed of $400 \mathrm{ft} / \mathrm{s}$ perpendicular to the wall.
Add the given vectors by drawing the appropriate resultant.(FIGURE CANNOT COPY)
Draw the given vectors and find their sum graphically. The magnitude is shown first, followed by the direction as an angle in standard position.$$3.6 \mathrm{cm}, 0^{\circ} ; 4.3 \mathrm{cm}, 90^{\circ}$$
Draw the given vectors and find their sum graphically. The magnitude is shown first, followed by the direction as an angle in standard position.$$2.3 \mathrm{cm}, 45^{\circ} ; 5.2 \mathrm{cm}, 120^{\circ}$$
Draw the given vectors and find their sum graphically. The magnitude is shown first, followed by the direction as an angle in standard position.$$6.0 \mathrm{cm}, 150^{\circ} ; 1.8 \mathrm{cm}, 315^{\circ}$$
Draw the given vectors and find their sum graphically. The magnitude is shown first, followed by the direction as an angle in standard position.$$7.5 \mathrm{cm}, 240^{\circ} ; 2.3 \mathrm{cm}, 30^{\circ}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{A}+\mathbf{B}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{B}+\mathbf{C}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{C}+\mathbf{D}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{D}+\mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{A}+\mathbf{C}+\mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{B}+\mathbf{D}+\mathbf{A}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$2 \mathbf{A}+\mathbf{D}+\mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{B}+\mathbf{E}+\mathbf{A}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{B}+3 \mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{A}+2 \mathbf{C}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$3 \mathbf{C}+\mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$2 \mathbf{D}+\mathbf{A}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{A}-\mathbf{B}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{C}-\mathbf{D}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{E}-\mathbf{B}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{C}-2 \mathbf{A}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$3 \mathbf{B}+\frac{1}{2} \mathbf{A}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$2 \mathbf{B}-\frac{3}{2} \mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{B}+2 \mathbf{C}-\mathbf{E}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{A}+2 \mathbf{D}-3 \mathbf{B}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{C}-\mathbf{B}-\frac{3}{4} \mathbf{A}$$
Find the indicated vector sums and differences with the given vectors by means of diagrams. (You might find graph paper to be helpful.)(IMAGE CANNOT COPY)$$\mathbf{D}-2 \mathbf{C}-\frac{1}{2} \mathbf{E}$$
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.Two forces that act on an airplane wing are called the lift and the drag. Find the resultant of these forces acting on the airplane wing in Fig. 9.10(FIGURE CANNOT COPY)
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.Two electric charges create an electric field intensity, a vector quantity, at a given point. The field intensity is $30 \mathrm{kN} / \mathrm{C}$ to the right and $60 \mathrm{kN} / \mathrm{C}$ at an angle of $45^{\circ}$ above the horizontal to the right. Find the resultant electric field intensity at this point.
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.A ski tow is moving skiers vertically upward at $24 \mathrm{m} / \mathrm{min}$ and horizontally at $44 \mathrm{m} / \mathrm{min}$. What is the velocity of the tow?
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.A jet travels $17 \mathrm{km}$ in a straight line as it also descends $8 \mathrm{km} .$ It then turns upward and travels $10 \mathrm{km}$ in a straight line until it reaches its original altitude, all in an easterly direction. What is the jet's displacement from its original position?
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.A driver takes the wrong road at an intersection and travels $4 \mathrm{mi}$ north, then 6 mi east, and finally 10 mi to the southeast to reach the home of a friend. What is the displacement of the friend's home from the intersection?
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.A ship travels $20 \mathrm{km}$ in a direction of $30^{\circ}$ south of east, then turns and goes due south for another $40 \mathrm{km},$ and finally turns again $30^{\circ}$ south of east and goes another $20 \mathrm{km} .$ What is the ship's displacement from its original position?
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.A rope holds a helium-filled balloon in place with a tension of $510 \mathrm{N}$ at an angle of $80^{\circ}$ with the ground due to a wind. The weight (a vertical force) of the balloon and contents is $400 \mathrm{N},$ and the upward buoyant force is $900 \mathrm{N}$. The wind creates a horizontal force of $90 \mathrm{N}$ on the balloon. What is the resultant force on the balloon?
Solve the given problems. Use a ruler and protractor as in Examples 6 and 7.While unloading a crate weighing $610 \mathrm{N},$ the chain from a crane supports it with a force of $650 \mathrm{N}$ at an angle of $20^{\circ}$ from the vertical. What force must a horizontal rope exert on the crate so that the total force (including its weight) on the crate is zero?