802 CHAPTER 8 Applications of Trigonometry Two vectors are equal if and only if they have the same direction and the same magnitude. In Figure 21, vectors A and B are equal, as are vectors C and D. As Figure 21 shows, equal vectors need not coincide, but they must be parallel and in the same direction. Vectors A and E are unequal because they do not have the same direction, while A≠F because they have different magnitudes. A B C D E F Figure 21 The sum of two vectors is also a vector. There are two ways to find the sum of two vectors A and B geometrically. 1. Place the initial point of vector B at the terminal point of vector A, as shown in Figure 22(a). The vector with the same initial point as A and the same terminal point as B is the sum A+ B. 2. Use the parallelogram rule. Place vectors A and B so that their initial points coincide, as in Figure 22(b). Then, complete a parallelogram that has A and B as two sides. The diagonal of the parallelogram with the same initial point as A and B is the sum A+ B. Parallelograms can be used to show that vector B + A is the same as vec- tor A+ B, or that A+ B = B + A, so vector addition is commutative. The vector sum A+ B is the resultant of vectors A and B. For every vector v there is a vector -v that has the same magnitude as v but opposite direction. Vector -v is the opposite of v. See Figure 23. The sum of v and -v has magnitude 0 and is the zero vector. As with real numbers, to subtract vector B from vector A, find the vector sum A+ 1-B2. See Figure 24. B A A + B B A A + B (b) (a) Figure 22 –v v Vectors v and –v are opposites. Figure 23 A –B A + (–B) B Figure 24 u 2u –2u u u 3 2 1 2 Figure 25 The product of a real number (or scalar) k and a vector u is the vector k # u, which has magnitude k times the magnitude of u. The vector k # u has the same direction as u if k 70 and the opposite direction if k 60. See Figure 25. The following properties are helpful when solving vector applications. Geometric Properties of Parallelograms 1. A parallelogram is a quadrilateral whose opposite sides are parallel. 2. The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a parallelogram are supplementary. 3. The diagonals of a parallelogram bisect each other, but they do not necessarily bisect the angles of the parallelogram.
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