801 8.3 Geometrically Defined Vectors and Applications 9. Volcano Movement To help predict eruptions from the volcano Mauna Loa on the island of Hawaii, scientists keep track of the volcano’s movement by using a “super triangle” with vertices on the three volcanoes shown on the map at the right. Find BC given that AB = 22.47928 mi, AC = 28.14276 mi, and A = 58.56989°. 10. Distance between Two Towns To find the distance between two small towns, an electronic distance measuring (EDM) instrument is placed on a hill from which both towns are visible. The distance to each town from the EDM and the angle between the two lines of sight are measured. See the figure. Find the distance between the towns. Mauna Loa Mauna Kea Hualalai A B C Town A Town B 43.33° 3428 m 5631 m Hill 8.3 Geometrically Defined Vectors and Applications ■ Basic Terminology ■ The Equilibrant ■ Incline Applications ■ Navigation Applications Basic Terminology Quantities that involve magnitudes, such as 45 lb or 60 mph, can be represented by real numbers called scalars. Other quantities, called vector quantities, involve both magnitude and direction. Typical vector quantities are velocity, acceleration, and force. For example, traveling 50 mph east represents a vector quantity. A vector quantity can be represented with a directed line segment (a segment that uses an arrowhead to indicate direction) called a vector. The length of the vector represents the magnitude of the vector quantity. The direction of the vector, indicated by the arrowhead, represents the direction of the quantity. See Figure 19. 10 lb Horizontal This vector represents a force of 10 lb applied at an angle 30° above the horizontal. 308 Figure 19 P O Vector OP, or vector u u P O Vector PO, or vector A A Vectors may be named with two uppercase letters or with one lowercase or uppercase letter. Figure 20 When we indicate vectors in print, it is customary to use boldface type or an arrow over the letter or letters. Thus, OP and OP> both represent the vector OP. When two letters name a vector, the first indicates the initial point and the second indicates the terminal point of the vector. Knowing these points gives the direction of the vector. For example, vectors OP and PO in Figure 20 are not the same vector. They have the same magnitude but opposite directions. The magnitude of vector OP is written ∣ OP∣.
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