# 11.2 Vector addition and subtraction: graphical methods

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• Understand the rules of vector addition, subtraction, and multiplication.
• Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects. 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 $\text{D}$ , stands for a vector. Its magnitude is represented by the symbol in italics, $D$ , and its direction by $\theta$ .

## 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 $\text{F}$ , which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as $F$ , and the direction of the variable will be given by an angle $\theta$ . A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle 29 .1º size 12{"29" "." "1º"} {} north of east. 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 29.1º size 12{"29" "." 1 rSup { size 12{º} } } {} north of east.

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. Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in [link] . (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or resultant vector     D size 12{A} {} . The length of the arrow D size 12{A} {} is proportional to the vector’s magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) θ size 12{θ} {} is measured with a protractor to be 29 . 1º size 12{"29" "." 1°} {} .

Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
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Rafiq
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LITNING
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Anam
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brayan
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Damian
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what king of growth are you checking .?
Renato
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
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research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Loga
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Bharti
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