2. Unit Vectors - Introduction The study of physics is...

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Unformatted text preview: Introduction The study of physics is intimately tied in with the study of mathematics. Sometimes, the direction of a number or quantity is as important as the number itself. Mathematicians in the 19th century developed a convenient way of describing and interacting with quantities with and without direction by dividing them into two types: scalar quantities and vector quantities. Scalar quantities have a magnitude but no direction. Vector quantities have both a magnitude and a direction. For instance, one might describe a plane as flying at 400 miles per hour. However, simply knowing the speed of the airplane is not nearly as useful as knowing the speed and direction of the airplane, so a more accurate description may be a plane flying at 400 miles per hour southeast. Scalar Quantities Scalar quantities are numbers that have a magnitude but no direction. Scalars are represented by a single letter, such as a. Some examples of scalar quantities are numbers without units (such as three), mass (five kilograms), and temperature (twenty- two degrees Celsius). Vector Quantities Vectors are a geometric way of representing quantities that have direction as well as magnitude. An example of a vector is force. If we are to fully describe a force on an object we need to specify not only how much force is applied, but also in which direction. Another example of a vector quantity is velocity -- an object that is traveling at ten meters per second to the east has a different velocity than an object that is traveling ten meters per second to the west. This vector is a special case; however, sometimes people are interested in only the magnitude of the velocity of an object. This quantity, a scalar, is called speed which has magnitude but no given direction. When vectors are written, they are represented by a single letter in bold type or with an arrow above the letter, such as or . Some examples of vectors are displacement (e.g. 120 cm at 30) and velocity (e.g. 12 meters per second north). The only basic SI unit that is a vector is the meter. All others are scalars. Derived quantities can be vector or scalar, but every vector quantity must involve meters in its definition and unit. Unit Vectors A unit vector has a magnitude of 1 with no units which is used to describe a point in space. It provides a convenient notation for expressions involving vector components. It is designated by symbol hat ^. A component vector is still a vector y x A A A + = A unit vector is used to specify a direction z k y j x i i A A x x = j A A y y = Then j A i A A y x + = Consider 2 vectors Vector B is expressed by j B i B B y x + = Then B A C + = ) ( ) ( j B i B j A i A C y x y x + + + = j B A i B A C y y x x ) ( ) ( + + + = So x x x B A C + = y y y B A C + = if the vector do not lie in the xy-plane, then a third component is introduced,...
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This note was uploaded on 05/19/2011 for the course PHYSICS 10 taught by Professor Darp during the Spring '11 term at Mapúa Institute of Technology.

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2. Unit Vectors - Introduction The study of physics is...

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