SECTION 9.4 Vectors 637 9.4 Vectors OBJECTIVES 1 Graph Vectors (p. 639) 2 Find a Position Vector (p. 640) 3 Add and Subtract Vectors Algebraically (p. 641) 4 Find a Scalar Multiple and the Magnitude of a Vector (p. 642) 5 Find a Unit Vector (p. 642) 6 Find a Vector from Its Direction and Magnitude (p. 643) 7 Model with Vectors (p. 644) Vectors are used in many different applications such as physics, engineering, and computer graphics, just to name a few. Vectors are also used in calculus as a different way to describe curves in the plane and in space. In simple terms, a vector (derived from the Latin vehere , meaning “to carry”) is a quantity that has both magnitude and direction. It is customary to represent a vector by using an arrow. The length of the arrow represents the magnitude of the vector, and the arrowhead indicates the direction of the vector. Many quantities in physics can be represented by vectors. For example, the velocity of an aircraft can be represented by an arrow that points in the direction of movement; the length of the arrow represents the speed. If the aircraft speeds up, we lengthen the arrow; if the aircraft changes direction, we introduce an arrow in the new direction. See Figure 45. Based on this representation, it is not surprising that vectors and directed line segments are somehow related. Geometric Vectors If P and Q are two distinct points in the xy -plane, there is exactly one line containing both P and Q [Figure 46(a)]. The points on that part of the line that joins P to Q , including P and Q , form what is called the line segment PQ [Figure 46(b)]. Ordering the points so that they proceed from P to Qresults in a directed line segment from P to Q , or a geometric vector , denoted by PQ. In a directed line segment PQ, P is called the initial point and Q the terminal point , as indicated in Figure 46(c). Figure 45 Figure 46 P Q (a) Line containing P and Q (b) Line segment PQ P Q Terminal point Initial point (c) Directed line segment PQ P Q The magnitude of the directed line segment PQ is the distance from the point P to the point Q ; that is, it is the length of the line segment. The direction of PQ is from P to Q . If a vector v has the same magnitude and the same direction as the directed line segment PQ, then = PQ v The vector v whose magnitude is 0 is called the zero vector , 0 . The zero vector is assigned no direction. Two vectors v and w are equal , written = v w if they have the same magnitude and the same direction. For example, the three vectors shown in Figure 47 have the same magnitude and the same direction, so they are equal, even though they have different initial points and different terminal points. As a result, it is useful to think of a vector simply as an arrow, keeping in mind that two arrows (vectors) are equal if they have the same direction and the same magnitude (length). NOTE In print, boldface letters are used to denote vectors, to distinguish them from numbers. For handwritten work, an arrow is placed over a letter to denote a vector. j Figure 47 Equal vectors R S P Q T U
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