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3.2 Vector addition and subtraction: graphical methods

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Learning objectives

By the end of this section, you will be able to:

  • Understand the rules of vector addition, subtraction, and multiplication.
  • Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)
  • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1)
Some Hawaiian Islands like Kauai Oahu, Molokai, Lanai, Maui, Kahoolawe, and Hawaii are shown. On the scale map of Hawaiian Islands the path of a journey is shown moving from Hawaii to Molokai. The path of the journey is turning at different angles and finally reaching its destination. The displacement of the journey is shown with the help of a straight line connecting its starting point and the destination.
Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai'i to Moloka'i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey)

Vectors in two dimensions

A vector    is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector's magnitude and pointing in the direction of the vector.

[link] shows such a graphical representation of a vector , using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction . We shall use the notation that a boldface symbol, such as D size 12{D} {} , stands for a vector. Its magnitude is represented by the symbol in italics, D size 12{D} {} , and its direction by θ size 12{θ} {} .

Vectors in this text

In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector F size 12{F} {} , which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as F size 12{F} {} , and the direction of the variable will be given by an angle θ size 12{θ} {} .

A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.
A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle 29 .1° size 12{"29" "." "1°"} {} north of east.
On a graph a vector is shown. It is inclined at an angle theta equal to twenty nine point one degrees above the positive x axis. A protractor is shown to the right of the x axis to measure the angle. A ruler is also shown parallel to the vector to measure its length. The ruler shows that the length of the vector is ten point three units.
To describe the resultant vector for the person walking in a city considered in [link] graphically, draw an arrow to represent the total displacement vector D size 12{D} {} . Using a protractor, draw a line at an angle θ size 12{θ} {} relative to the east-west axis. The length D size 12{D} {} of the arrow is proportional to the vector's magnitude and is measured along the line with a ruler. In this example, the magnitude D size 12{D} {} of the vector is 10.3 units, and the direction θ size 12{θ} {} is 29.1° size 12{"29" "." 1 rSup { size 12{°} } } {} north of east.

Vector addition: head-to-tail method

The head-to-tail method    is a graphical way to add vectors, described in [link] below and in the steps following. The tail    of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.

Questions & Answers

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In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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