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Unformatted text preview: Physical Sciences 2 Fall 2010 Vectors and Motion Vectors are covered in Chapter 1 of Bauer and Westfall, and this handout will provide an overview of vector notation and some additional topics about vectors not covered in Chapter 1. If you’ve never seen vectors at all, you should definitely read section 16 of Bauer and Westfall before continuing with this handout. Notation We’ll use arrowheads to denote any vector quantity. The key is consistency: literally every symbol with an arrowhead over it refers to a vector, and almost anything without an arrowhead symbol is not a vector. Why “almost”? Well, there are two exceptions. The first is that the zero vector is generally written as 0 instead of . The second is, by convention, unit vectors (special vectors with a magnitude of 1 that point in the directions of the coordinate axes) are represented with little hats instead of arrowheads: ˆ x = unit vector pointing in xdirection ˆ y = unit vector pointing in ydirection ˆ z = unit vector pointing in zdirection If it helps, you can think of the hats as a special version of the arrowheads. But there are many scalar quantities that are related to vectors, but generally are not written with arrowheads: 1. Magnitude . If v is a vector, we’ll often want to talk about the magnitude of v , which is a scalar. The clearest and most explicit way of doing so is to use vertical bars for magnitude, just like absolute value: v = some vector v = magnitude of v (a scalar) However, it is also very common to denote the magnitude of v simply with the letter v (with no arrowhead). You should be comfortable with this notation; in particular, note that v is a scalar (no arrowhead), but it is a scalar that is closely related to the vector v . Also, remember that v is always a positive number (or zero)—the magnitude of a vector cannot be negative. 2. Components . Given a vector and a coordinate system, we can decompose the vector into its components along the coordinate axes, as described in Section 16 of Bauer and Westfall. For example, suppose vector a has a magnitude of 10 and points 30° above the positive xaxis, as in Figure 1 at right. Then we can consider the horizontal and vertical parts of a separately. The horizontal component is labeled a x and the vertical component is denoted a y . (If this were a threedimensional problem, we could also have a z component a z .) The notation is important: we use the letter that names the vector and indicate which component by using an x , y , or z subscript. Physical Sciences 2 Fall 2010 Using trigonometry, we can determine the numerical values of the components: a x = 10cos30 ° = 5 3 ≈ 8.7 a y = 10sin 30 ° = 5 Going the other way (from components to magnitude and direction) isn’t hard either: a = a x 2 + a y 2 = 5 3 ( ) 2 + 5 ( ) 2 = 10 θ = arctan a y a x = arctan 1 3 = 30 ° There are some very important things to keep in mind when dealing with components: a. a....
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