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Unformatted text preview: (8/1/08) Math 20C. Lecture Examples. Section 12.1. Vectors in the plane † Definition 1 A vector v represents a nonnegative number and, if the number is not zero, a direction. The number associated with the vector v is called its length or magnitude and is denoted | v | . The vector of zero length is the zero vector and is denoted O . It has no direction. Vectors are generally denoted by bold-faced letters, like the v in Definition 1, or by letters with arrows over them, as in the symbol-→ PQ . Vectors are represented in drawings by arrows, where in each case the length of the arrow is the magnitude of the vector, measured with a scale that might or might not be the scale used on the coordinate axes. The direction of the arrow is the direction associated with the vector (Figure 1). The same vector can be represented by different arrows in different locations (Figure 2), provided that the different arrows are parallel, have the same lengths, and point in the same direction. v | v | v v FIGURE 1 FIGURE 2 If a vector is represented by an arrow in an xy-plane as in Figure 3, then the x- and y-components of the vector are the changes in the x- and y-coordinates from the base to the tip of the arrow, measured with the scale used for measuring the length of the arrow. If the x-component of v is a and the y-component is b , as shown in Figure 6 with positive a and b , we write v = ( a,b ) . The length of a vector can be calculated from its x- and y-components by using the Pythagorean Theorem: | v | = |( a,b )| = radicalbig a 2 + b 2 . x y v = ( a,b ) a b FIGURE 3 † Lecture notes to accompany Section 12.1 of Calculus, Early Transcendentals by Rogawski. 1 Math 20C. Lecture Examples. (8/1/08) Section 12.1, p. 2 A nonzero vector v in an xy-plane can also be described by giving its length | v | and its angle of inclination , which is an angle θ...
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