Work A heavy object is pulled 30 feet across a floor, using a force of 100 pounds. The

Fill in the blanks. A ________ ________ ________ can be used to represent a quantity that involves both magnitude and direction.

Fill in the blanks. The directed line segment has ________ point and ________ point

Fill in the blanks. The ________ of the directed line segment is denoted by

Fill in the blanks. The set of all directed line segments that are equivalent to a given directed line segment is a ________ in the plane.

Fill in the blanks. In order to show that two vectors are equivalent, you must show that they have the same ________ and the same ________ .

Fill in the blanks. The directed line segment whose initial point is the origin is said to be in ________ ________ .

Fill in the blanks. A vector that has a magnitude of 1 is called a ________ ________ .

Fill in the blanks. The two basic vector operations are scalar ________ and vector ________ .

Fill in the blanks. The vector is called the ________ of vector addition.

Fill in the blanks. The vector sum is called a ________ ________ of the vectors and and the scalars and are called the ________ and ________ components of respectively.

Showing That Two Vectors Are Equivalent In Exercises 11 and 12, show that u and v are equivalent.

Showing That Two Vectors Are Equivalent In Exercises 11 and 12, show that u and v are equivalent.

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Finding the Component Form of a Vector In Exercises 1324, find the component form and magnitude of the vector v

Sketching the Graph of a Vector In Exercises 2530, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.

Sketching the Graph of a Vector In Exercises 2530, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.

Sketching the Graph of a Vector In Exercises 2530, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.

Sketching the Graph of a Vector In Exercises 2530, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.

Sketching the Graph of a Vector In Exercises 2530, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.

Sketching the Graph of a Vector In Exercises 2530, use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Vector Operations In Exercises 3138, find (a) (b) and (c) Then sketch each resultant vector.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Unit Vector In Exercises 39 48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.

Finding a Vector In Exercises 4952, find the vector v with the given magnitude and the same direction as u.

Finding a Vector In Exercises 4952, find the vector v with the given magnitude and the same direction as u.

Finding a Vector In Exercises 4952, find the vector v with the given magnitude and the same direction as u.

Finding a Vector In Exercises 4952, find the vector v with the given magnitude and the same direction as u.

Writing a Linear Combination of Unit Vectors In Exercises 5356, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors i and j.

Writing a Linear Combination of Unit Vectors In Exercises 5356, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors i and j.

Writing a Linear Combination of Unit Vectors In Exercises 5356, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors i and j.

Writing a Linear Combination of Unit Vectors In Exercises 5356, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors i and j.

56. Vector Operations In Exercises 5762, find the component form of v and sketch the specified vector operations geometrically, where

56. Vector Operations In Exercises 5762, find the component form of v and sketch the specified vector operations geometrically, where

56. Vector Operations In Exercises 5762, find the component form of v and sketch the specified vector operations geometrically, where

56. Vector Operations In Exercises 5762, find the component form of v and sketch the specified vector operations geometrically, where

56. Vector Operations In Exercises 5762, find the component form of v and sketch the specified vector operations geometrically, where

56. Vector Operations In Exercises 5762, find the component form of v and sketch the specified vector operations geometrically, where

Finding the Direction Angle of a Vector In Exercises 6366, find the magnitude and direction angle of the vector v.

Finding the Direction Angle of a Vector In Exercises 6366, find the magnitude and direction angle of the vector v.

Finding the Direction Angle of a Vector In Exercises 6366, find the magnitude and direction angle of the vector v.

Finding the Direction Angle of a Vector In Exercises 6366, find the magnitude and direction angle of the vector v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 6774, find the component form of v given its magnitude and the angle it makes with the positive -axis. Sketch v.

Finding the Component Form of a Vector In Exercises 7578, find the component form of the sum of u and v with direction angles and

Finding the Component Form of a Vector In Exercises 7578, find the component form of the sum of u and v with direction angles and

Finding the Component Form of a Vector In Exercises 7578, find the component form of the sum of u and v with direction angles and

Finding the Component Form of a Vector In Exercises 7578, find the component form of the sum of u and v with direction angles and

Using the Law of Cosines In Exercises 79 and 80, use the Law of Cosines to find the angle between the vectors. (Assume )

Using the Law of Cosines In Exercises 79 and 80, use the Law of Cosines to find the angle between the vectors. (Assume )

Resultant Force In Exercises 81 and 82, find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive -axis and force 2 as a vector at an angle with the positive -axis.)

Resultant Force In Exercises 81 and 82, find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive -axis and force 2 as a vector at an angle with the positive -axis.)

Velocity A gun with a muzzle velocity of 1200 feet per second is fired at an angle of above the horizontal. Find the vertical and horizontal components of the velocity.

Velocity Pitcher Joel Zumaya was recorded throwing a pitch at a velocity of 104 miles per hour. Assuming he threw the pitch at an angle of below the horizontal, find the vertical and horizontal components of the velocity. (Source: Damon Lichtenwalner, Baseball Info Solutions)

Resultant Force Forces with magnitudes of 125 newtons and 300 newtons act on a hook (see figure). The angle between the two forces is Find the direction and magnitude of the resultant of these forces. Figure for 85 Figure for 86

Resultant Force Forces with magnitudes of 2000 newtons and 900 newtons act on a machine part at angles of and respectively, with the -axis (see figure). Find the direction and magnitude of the resultant of these forces.

Resultant Force Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of and respectively, with the positive -axis. Find the direction and magnitude of the resultant of these forces.

Resultant Force Three forces with magnitudes of 70 pounds, 40 pounds, and 60 pounds act on an object at angles of and respectively, with the positive -axis. Find the direction and magnitude of the resultant of these forces.

Cable Tension The cranes shown in the figure are lifting an object that weighs 20,240 pounds. Find the tension in the cable of each crane.

Cable Tension Repeat Exercise 89 for and

Cable Tension In Exercises 91 and 92, use the figure to determine the tension in each cable supporting the load.

Cable Tension In Exercises 91 and 92, use the figure to determine the tension in each cable supporting the load.

Tow Line Tension A loaded barge is being towed by two tugboats, and the magnitude of the resultant is 6000 pounds directed along the axis of the barge (see figure). Find the tension in the tow lines when they each make an angle with the axis of the barge.

Rope Tension To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are tied to an eyelet on the top center of the cylinder. Each rope makes a angle with the vertical. Draw a figure that gives a visual representation of the situation. Then find the tension in the ropes.

Find when pounds and

Find when pounds and

Find when pounds and pounds.

Find when pounds and

Work A heavy object is pulled 30 feet across a floor, using a force of 100 pounds. The force is exerted at an angle of above the horizontal (see figure). Find the work done. (Use the formula for work, where is the component of the force in the direction of motion and is the distance.)

Rope Tension A tetherball weighing 1 pound is pulled outward from the pole by a horizontal force until the rope makes a angle with the pole (see figure). Determine the resulting tension in the rope and the magnitude of u 1 lb Tension 45 u.

Navigation An airplane is flying in the direction of with an airspeed of 875 kilometers per hour. Because of the wind, its groundspeed and direction are 800 kilometers per hour and respectively (see figure). Find the direction and speed of the wind.

Navigation A commercial jet is flying from Miami to Seattle. The jets velocity with respect to the air is 580 miles per hour, and its bearing is The wind, at the altitude of the plane, is blowing from the southwest with a velocity of 60 miles per hour. (a) Draw a figure that gives a visual representation of the situation. (b) Write the velocity of the wind as a vector in component form. (c) Write the velocity of the jet relative to the air in component form. (d) What is the speed of the jet with respect to the ground? (e) What is the true direction of the jet?

True or False? In Exercises 103106, determine whether the statement is true or false. Justify your answer. If and have the same magnitude and direction, then and are equivalent.

True or False? In Exercises 103106, determine whether the statement is true or false. Justify your answer. If is a unit vector in the direction of then

True or False? In Exercises 103106, determine whether the statement is true or false. Justify your answer. f then

True or False? In Exercises 103106, determine whether the statement is true or false. Justify your answer. If is a unit vector, then

Proof Prove that is a unit vector for any value of

Technology Write a program for your graphing utility that graphs two vectors and their difference given the vectors in component form.

Finding the Difference of Two Vectors In Exercises 109 and 110, use the program in Exercise 108 to find the difference of the vectors shown in the figure.

Finding the Difference of Two Vectors In Exercises 109 and 110, use the program in Exercise 108 to find the difference of the vectors shown in the figure.

Graphical Reasoning Consider two forces (a) Find as a function of (b) Use a graphing utility to graph the function in part (a) for (c) Use the graph in part (b) to determine the range of the function. What is its maximum, and for what value of does it occur? What is its minimum, and for what value of does it occur? (d) Explain why the magnitude of the resultant is never 0.

HOW DO YOU SEE IT? Use the figure to determine whether each statement is true or false. Justify your answer.

Writing Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar.

Writing Identify the quantity as a scalar or as a vector. Explain your reasoning. (a) The muzzle velocity of a bullet (b) The price of a companys stock (c) The air temperature in a room (d) The weight of an automobile