Vectors and Index Notation - UCA

Vectors and Index NotationStephen R. AddisonJanuary 12, 20041 Basic vector Unit VectorsWe will denote a unit vector with a superscript caret, thus adenotes a unit vector . a | a|=1If~xis a vector in thex-direction x=~x|~x|is a unit vector . We will usei,j,and k, or x, y,and z, ore1,e2and e3and a variety of variations without further Addition and SubtractionAddition and subtraction are depicted Scalar Products~a.~b=|~a||~b|cos 1~a.~b=~b.~asince cos =cos( ) vector Products|~a ~b|=|~a||~b|sin ~a ~b= ~b ~a,Why?~c=~a ~b,~cis perpendicular to~aand~ Geometric InterpretationQPqArea of a triangle=12base perpendicular height=12|~Q||~P|sin SoA=12|~Q ~P|is the area of a triangle and, accordingly,|~Q ~P|=the area of a Area as a vectorAreas can also be expressed as vector quantities, for the parallelogram considered above, we couldhave written~A=~Q ~P.

The unit vectors i, j,and k are a basis of R3. So we will often denote ~A as Ai with the understanding that the index can assume the values 1, 2, or 3 independently. Ai stands for the scalar components (A1,A2,A3); we’ll refer to the vector Ai even though to get ~A we need to calculate Aiei. 2.1.1 Examples ai =bi ⇒ a1 =b1, a2 =b2, a3 =b3 ...

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