ch1b - Notes on Chapter 1 (pp. 11-26) Vectors To describe...

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John Ellison, UCR p.1 Notes on Chapter 1 (pp. 11-26) Vectors To describe motion in three dimensions we will use vectors . A vector has magnitude and direction. An example of a vector quantity is displacement. The displacement of a particle which moves from point A to point B can be represented by a vector. Its magnitude is the length of the line joining A and B, and its direction is from A to B, indicated by an arrow. Other examples of vector quantities are velocity and acceleration. A scalar is a quantity which has magnitude only (no direction). Examples of scalar quantities are speed, temperature, mass, time, and pressure. We will represent vectors by italic symbols with an arrow above, e.g. . The magnitude of a vector is represented by the symbol with no arrow, e.g. a , or by the notation . Now consider two displacements, one from A to B, and the other from B to C. The net displacement is from A to C. This can be represented graphically as shown in the figure at top right. We call AC the vector sum (or resultant ) of the vectors AB and AC. We can represent this by the vector addition equation which says that vector is the vector sum of vectors and . Note that this is not a simple algebraic addition - it involves the magnitudes and directions of the vectors, not just the magnitudes. Vector addition has two important properties: Vector subtraction is defined by: where the vector is a vector with the same magnitude as but with the opposite direction (see the figure below). s = a b a b = b  a (commutatative law)  a b  c =  a  b c (associative law) a d a b a − b s a b ∣ a b b
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John Ellison, UCR p.2 Unit Vectors The component of a vector is the projection of the vector on an axis (see figure below). The x component and y component are the projections on the x -axis and y -axis respectively, and are denoted by a x and a y , respectively. The process of finding the components of a vector is called resolving the vector . From simple geometry (see the figure below), we find that the components are given by: where θ is the angle the vector makes with the positive direction of the x -axis. In order to express a vector in terms of its components we define unit vectors as vectors of unit magnitude which point in the direction of the positive x , y , z directions, respectively. We will normally use a coordinate system in which the + x direction is horizontal, the + y direction is vertical, and the + z direction is "out of the page". Such a coordinate system (shown in the figure above left) is said to be a right-handed coordinate system . Now we can express the vectors as This says that vector is equal to the sum of a vector of magnitude a x along the + x direction and a vector of magnitude a y along the + y direction (and similarly for ). The magnitude and angle of vector are a = a x i a y j b = b x i b y j a a and b b a x = a cos and a y = a sin magnitude: a = a x 2 a y 2
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ch1b - Notes on Chapter 1 (pp. 11-26) Vectors To describe...

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