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Unformatted text preview: 1,... u. -.,. my: "'FZ— .t. «ml 2:! fP‘Wlfil-I‘; 1'1" 32 Chapter 3 Motion in Two and Three Dimensions HERB} Vectors i'} and A? suin to 7;. D 1,. 30° The vector Fl describes the position of this point. 'MUFHWQ 0t Minn . “Dot some: ” A? is a vector quantity, and it’s crucial to include the arrow whenever you’re dealing with vec- tors. Figure 3.1 shows a position vector in a two-dimensional coordinate system; this vec- tor describes a point a distance of 2 m from the origin, in a direction 30° from the horizontal axis. Suppose you walk from the origin straight to the point described by the vector F1 in Fig. 3.1, and then you turn right and walk another 1 m. Figure 3.2 shows how you can tell where you end up. Draw a second vector whose length represents 1 m and that points to the right; we’ll call this vector A? because it’s a displacement vector, representing a change in position. Put the tail of A? at the head of the vector 71; then the head of A? shows your ending position. The result is the same as if you had walked straight from the origin to this position. 50 the new position is described by a third vector F2, as indicated in Fig. 3.2. What we’ve just described is the process of vector addition. To add two vectors, put the second vector’s tail at the head of the first; the sum is then the vector that extends from the tail of the first vector to the head of the second, as does F2 in Fig. 3.2. Remember that a vector has both magnitude and direction—but because that’s all the information it contains, it doesn’t matter where a vector starts. So you’re free to move a vector around anywhere as needed to form vector sums. Figure 3.3 shows some examples of vector addition and also shows that vector addition obeys simple rules you know for reg— ular arithmetic. V d di . . . . Vectorfladdition is_ also associative: ectora mortars commutative. (A + B) + 5=A + (B + C). E+§=§+A r s 5+3 4 A r+§ A s W Vector addition is commutative and associative. Multiplication You and I both jog in the same direction, but you go twice as far. Your displacement vector, E, is then twice as long as my displacement vector, 2i; mathematically, we write 3 = 2/1. That’s what it means to multiply a vector by a scalar; simply rescale the magnitude of the vector by that scalar. If the scalar is negative, then the vector direction reverses-and that provides a way to subtract vectors. In Fig. 3.2, for example, you can see that 71 = F; + (-1)A?, or simply F1: 72 7 13?. Later, we’ll see ways to multiply two vectors, but for now the only multiplication we consider is a vector multiplied by a scalar. Vector Components You can always add vectors by the graphical method shown in Fig. 3.2, or you can use geo- metric relationships lilce the laws of sines and cosines to accomplish the same thing a1ge~ braically. In both these approaches, you specify a vector by giving its magnitude and its direction. But often it’s more convenient instead to describe vectors in terms of their components in a given coordinate system. A coordinate system is a framework for describing positions in space. It’s a mathemat— ical construct, and you’re free to choose whatever coordinate system you want. You‘ve al- ready seen Cartesian or rectangular coordinate systems, in which a pair of numbers (x, y) represents each point in a plane. You could also think of each point as representing the head of a position vector, in which case the numbers x and y are the vector components. The components tell how much of the vector is in the x direction and how much is in the y direction. Not all vectors represent actual positions in space; for example, there are ...
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