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Vectors

Vectors. A vector is a quantity or phenomenon that has two independent properties: magnitude and direction. The term also denotes the mathematical or geometrical representation of such a quantity.

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Vectors

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  1. Vectors A vector is a quantity or phenomenon that has two independent properties: magnitude and direction. The term also denotes the mathematical or geometrical representation of such a quantity. Examples of vectors in nature are velocity, momentum, force, electromagnetic fields, and weight. (Weight is the force produced by the acceleration of gravity acting on a mass.) A quantity or phenomenon that exhibits magnitude only, with no specific direction, is called ascalar. Examples of scalars include speed, mass, electrical resistance, and hard-drive storage capacity. • Scalars are quantities that are fully described by a magnitude (or numerical value) alone. • Vectors are quantities that are fully described by both a magnitude and a direction Vectors can be depicted graphically in two or three dimensions. Magnitude is shown as the length of a line segment. Direction is shown by the orientation of the line segment, and by an arrow at one end.

  2. Coordinate Systems In two dimensions, this description is accomplished with the use of the Cartesian coordinate system,in which perpendicular axes intersect at a point defined as the origin O (Fig. 3.1).Cartesian coordinates are also called rectangular coordinates. x =r cos q y =r sin q Figure 3.1 Designation of points in a Cartesian coordinate system. Every point is labeled with coordinates (x, y). Figure 3.2 (a) The plane polar coordinates of a point are represented by the distance r and theangle q, where u is measured counterclockwise from the positive x axis. (b) The right triangle used torelate (x, y) to (r,q).

  3. x =r cos q y =r sin q r= Example 3.1 The Cartesian coordinates of a point in the xyplane are (x, y) = (-3.50, -2.50) m. Find thepolar coordinates of this point.

  4. Vector and Scalar Quantities • A scalar quantity is completely specified by a single value with an appropriateunit and has no direction. Examplesof scalar quantities are volume, mass, speed, time, and time intervals.Some scalars are always positive, such as mass and speed. Others, such astemperature, can have either positive or negative values. The rules of ordinaryarithmetic are used to manipulate scalar quantities. • A vector quantity is completely specified by a number with an appropriateunit (the magnitude of the vector) plus a direction.has no direction.

  5. Adding Vectors + + Geometric construction for summing four vectors. The resultant vector R is by definition the one that completes the polygon.

  6. Negative of a Vector Subtracting Vectors The operation of vector subtraction makes use of the definition of the negative of avector. We define theoperation A-B as vector -B added to vector A The geometric construction for subtracting two vectors in this way is illustrated in Figure

  7. Example A car travels 20.0 km due north and then 35.0 kmin a direction 60.0° west of north.Find the magnitude and direction of the car’s resultant displacement. Solution The resultant displacement of the car is 48.2 km in a direction 38.9° west of north

  8. Components of a Vector and Unit Vectors In this section, we describe a method of adding vectors that makes use of the projections of vectorsalong coordinate axes. These projections are called the components of the vectoror its rectangular components. Any vector can be completely described by its components. The magnitudes of these components are the lengths of the two sides of a right trianglewith a hypotenuse of length A. Therefore, the magnitude and direction of A are related to its components through the expressions.

  9. Unit Vectors Unit vector is a dimensionless vector having a magnitude of exactly 1. Unit vectors are used to specifya given direction and have no other physical significance The symbols ,and to represent unit vectors pointing in the positive x, y, and zdirections, respectively. The magnitude of each unit vector equals 1; that is, = ==1 = = = +

  10. =+ = ( + ) + ( + or = ( + ) + ( + ) Because = + we see that the components of the resultant vector are The magnitude of and the angle it makes with the x axis are obtained from its components using the relationships R= tanq= =

  11. At times, we need to consider situations involving motion in three componentdirections. The extension of our methods to three-dimensional vectors is straightforward. If and both have x, y, and z components, they can be expressed in the form = + + = + + The sum of and is = ( + ) + ( + ) + ( + ) The angle that makes with the x axis is found from the expression ,with similarexpressions for the angles with respect to the y and z axes. cos=

  12. Find the magnitude of the resultant vector ?

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