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09 Vectors.pdf

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Douglas Tobias

Vectors: A scalar quantity: one that has a single value (magnitude) at a particular point in space. ex: distance, speed, electric potential. A vector: quantity has both magnitude and direction at each point in space. ex: position, velocity, electric field. the set of N vectors e1, e2, ..., n are said to be lienarly independent if N ⁄i=0a i i 0 Implies that a = a =...= a = 0 1 2 N In this case the vectors e ipan an N-dimensional vector space, and they can be chosen as the basis vectors in that space. An arbitrary vector v can therefore be expressed as v = N v e = v e + v e + ...+ v e ⁄i=1 i i 1 1 2 2 N n With not all the constants v iqual to zero. With the choice of basis vectors understood, a vector may be specified by listing its components v = Hv1, 2 , ...,Nv L In Mathematica, we specify vectors as lists of their components In[30]:[email protected], v, r, sD; u = 8u1, u2, u3< v = 8v1, v2, v3< r = 81, 1, 1< s = 8-1, -2, 1< Out[31]=u1, u2, u3< Out[32]=v1, v2, v3< Out[33]=1, 1, 1< Out[34]=-1, -2, 1< Addition and subtraction of vectors with the same basis vectors is performed by adding or subtracting their components Printed by Wolfram Mathematica Student Edition 2 Untitled-1 u + v In[35]:= Out[35]=u1 + v1, u2 + v2, u3 + v3< In[7]:=+ s Out[7]=0, -1, 2< In[8]:=- u Out[8]=-u1 + v1, -u2 + v2, -u3 + v3< In[9]:=- s Out[9]=2, 3, 0< Multiplication of a vector by a scalar, a, simply scales each of the components by a: In[10]:=* v Out[10]=a v1, a v2, a v3< In[11]:=* r Out[11]=5, 5, 5< In[12]:=ê 2 1 1 Out[12]=- , -1, > 2 2 There are two types of vector products. The scalar product, also known as the dot product is: In[13]:=v Out[13]=1 v1 + u2 v2 + u3 v3 In[14]:[email protected], vD Out[14]=1 v1 + u2 v2 + u3 v3 In[15]:[email protected], uD Out[15]=1 v1 + u2 v2 + u3 v3 In[16]:[email protected], sD Out[16]=2 The vector product, also known as the cross product is a vector quantity: In[17]:[email protected], vD Out[17]=-u3 v2 + u2 v3, u3 v1 - u1 v3, -u2 v1 + u1 v2< In[18]:[email protected], uD Out[18]=u3 v2 - u2 v3, -u3 v1 + u1 v3, u2 v1 - u1 v2< Printed by Wolfram Mathematica Student Edition Untitled-1 3 In[19]:[email protected], sD 83, -2, -1< Out[19]= Note: the dot product is commutative u ·v = v u the cross product is not, b/c u x v = -v x u. The magnitude (norm) of a vector is: In[20]:[email protected] Out[20]= [email protected] + [email protected] + [email protected] 2 In[21]:[email protected] Out[21]= 6 A unit vector is a vector with magnitude equal to 1. A unit vector in the direction of u is obtained by dividing each of the compoents by the magnitude u. The Normalize command dos this: In[22]:[email protected] v1 Out[22]= , 2 2 2 [email protected] + [email protected] + [email protected] v2 v3 , > 2 2 2 2 2 2 [email protected] + [email protected] +
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