Chapter 3: Section 3 Learning Goal: To introduce you to vectors and the use of sine and cosine for a triangle when resolving components. Vectors are an important part of the language of science, mathematics, and engineering. They are used to discuss multivariable calculus, electrical circuits with oscillating currents, stress and strain in structures and materials, and flows of atmospheres and fluids, and they have many other applications. Resolving a vector into components is a precursor to computing things with or about a vector quantity. Because position, velocity, acceleration, force, momentum, and angular momentum are all vector quantities, resolving vectors into components is the most important skill required in a mechanics course. The figure (Intro 1 figure) shows the components of , and , along the x and y axes of the coordinate system, respectively. The components of a vector depend on the coordinate system's orientation, the key being the angle between the vector and the coordinate axes, often designated . Part A: Q: The figure (Part A figure) shows the standard way of measuring the angle. is measured to the vector from the x axis, and counterclockwise is positive. A: , = Part B: Q: When you resolve a vector into components, the components must have the form or . The signs depend on which quadrant the vector lies in, and there will be one component with and the other with . In real problems the optimal coordinate system is often rotated so that the x axis is not horizontal. Furthermore, most vectors will not lie in the first quadrant. To assign the sine and cosine correctly for vectors at arbitrary angles, you must figure out which angle is and then properly reorient the definitional triangle. As an example, consider the vector shown in the diagram (Part B figure) labeled "tilted axes," where you know the angle between and the y axis. Which of the various ways of orienting the definitional triangle must be used to resolve into components in the tilted coordinate system shown? (In the figures, the hypotenuse is orange, the side adjacent to is red, and the side opposite is yellow.) A: 4 Part C: Q: Choose the correct procedure for determining the components of a vector in a given coordinate system from this list: A: Align the adjacent side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle. Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate direction with as the included angle. Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle. Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate direction with as the included angle. Part D: Q: The space around a coordinate system is conventionally divided into four numbered quadrants depending on the signs of the x and ycoordinates (Part D figure) . Consider the following conditions: A. , B. , C. , D. , Which of these conditions are true in which quadrants? A: 1,4,2,3 Part E: Q: Now find the components and of in the tilted coordinate system of Part B. A: , =
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