ME1302 - Part 3 - VECTORS AND VECTOR CALCULUS Basic...

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VECTORS AND VECTOR CALCULUS Basic Definitions A scalar is a quantity that can be completely described by means of a single number which measures its magnitude, in appropriate units. A vector is a quantity that needs to be specified by its magnitude and direction. Typical scalar quantities are temperature, mass, length, time and volume. Typical vector quantities are displacement, velocity, acceleration, force and heat flux. A vector is geometrically represented by a straight line segment, proportional to its magnitude, with an arrow to indicate its direction. A reversal of the arrow keeps the magnitude unchanged but reverses its direction. Two vectors are only equal if they have the same magnitude and direction. A vector remains unchanged if it is moved parallel to itself without any change of length or direction. Such a shift is called a translation . Notation A vector from the initial point A to the terminal point B is denoted by , or by a single bold face symbol like a . AB JJJG The magnitude of the vector AB is denoted by AB (or by | a |). This represents the length of the line connecting points A and B . A vector of unit length in the direction of a is called a unit vector and written as â . Thus, | â | = 1. Vector Addition T add vectors and CD , we connect the ends B and C which results in the vector JJJ . This is called the triangle law. Alternatively, we can use the parallelogram law which involves connect the initial points A and C and then completing the parallelogram. o AB AD G
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Addition Properties Commutative law: + =+ abba Associative law: ( ) ( ) + =++ ab + c abc If k and l are scalars, then ( ) kl k l +=+ aa a ( ) kk k += ab a + b ( ) ( ) ( ) kl lk k l == aa a If we set k = | a | and l = 1/| a |, the last equation becomes 1  =   a a But ˆ = a a a is the unit vector in the direction of a , so we arrive at the result ˆ = a The above equation expresses the fact that any vector a may be written as the product of the unit vector â scaled by the magnitude of a . Other Vector Operations Subtraction of Vectors The vector b is defined as having the same magnitude as b , but with the opposite direction. Thus, () =+− c=a b a b If a vector is represented as , then AB JJJG AB BA = JJJ GJ J J G .
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Scaling Vectors Scaling vectors means multiplying them by a scalar. The result of the product of a vector a by a scalar k is a vector with magnitude | k | | a |, with the same direction as a if k > 0 but with its direction reversed if k < 0. Vectors in Component Form If a point P has Cartesian coordinates P ( x , y , z ), it follows that the vector OP (where O (0,0,0)) can be represented in component form as JJJG OP x y z = ++ r= i j k The magnitude of the above vector is given by ( ) 1/2 222 rO P x y z == = ++ r If the angles between OP and the , and x yz -axes are given by , and α βγ , respectively, then if we set cos , cos , cos lmn === it follows that cos x rr l = = cos yr r m β = = cos zr r n γ = = Thus, ( ) x y z rl rm rn r l m n =++ + + ri j k= i j k= i j k and r = =+ + r j k ± The numbers (in this order) are called the direction cosines of vector r and,
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This note was uploaded on 04/20/2011 for the course ME 1331 taught by Professor Zeshan during the Spring '11 term at American College of Computer & Information Sciences.

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ME1302 - Part 3 - VECTORS AND VECTOR CALCULUS Basic...

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