vectoralgebra - 1. Vector Algebra 1.1 Review of Basic Ideas...

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1. Vector Algebra 1.1 Review of Basic Ideas (p.1 – p.8) In engineering and science, physical quantities which are completely specified by their magnitude (size) are known as scalars . Examples are: mass, temperature, volume, resistance, charge, voltage, current, etc. Other quantities possess both magnitude and direction are known as vectors . Examples of vector quantities are: velocity, acceleration, force, electric field, magnetic field etc and will be denoted by , etc. Vectors may be represented geometrically by directed line segments. If A and B are two geometrical points in , , , , vaFEB GGJ GJ G 3 R , the directed line segment from A to B is called the vector from A to B and is denoted AB JJJG . As the name implies, this is a vector quantity with direction from A to B and magnitude the distance between and B. The vector AB is represented by an arrow from A to B as shown in the following figure. The point is called the initial point of the vector AB , and B is called the terminal point of AB . The magnitude of the vector AB is called its length and is denoted AB . 1. Two vectors a and are G b G equal if they have the same magnitude and direction. We write ab = G G . Two vectors can be the same even though the initial points and terminal points are different. For example A BO P = in 2 R because they have the same length and the same direction (they both proceed one unit to the left and two units up). Thus the same vector can be translated from one position to another; what is important is that the length and direction remain the same, and not where the initial points and terminal points are located. For this reason, we shall often denote vectors as , etc. which make no reference to the initial points and terminal points. , , abc GGG 2. A vector having the same magnitude as a G but the opposite direction is denoted by . a G 1
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3. Geometrically the sum of two vectors is given by the parallelogram law of vector addition or tip - to - tail method of adding vectors. Parallelogram law of vector addition: In the parallelogram determined by two vectors , ab G G , the vector is the diagonal with the same initial point as + GG G G . Tip - to - tail method of adding vectors: Given two vectors G G , place the tail of at the tip of b G a G . Then is the vector from the tail of abc += GGG a G to the tip of b G . 4. The difference of two vectors a G and b G represented by cab = G is defined as ( ) . ca b =+− G 5. If a then a is the b = b zero vector denoted by0 G . This has magnitude 0 but no direction. 6. Multiplication of a by a scalar, m , produces a vector G ma G with magnitude m times that of a G and direction the same as or opposite to that of a G according to whether m is positive or negative respectively. If then . 0 m = 0 = 2
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7. Unit vectors are vectors with magnitude 1. If a G is any vector then we usually denote its magnitude by a G . A unit vector with the same direction as ( ) 0 a G G will be a a G G .
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vectoralgebra - 1. Vector Algebra 1.1 Review of Basic Ideas...

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