Engineering Calculus Notes 39

Engineering Calculus Notes 39 - + y + z k . We shall Fnd it...

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1.2. VECTORS AND THEIR ARITHMETIC 27 that if −→ v = ( x,y,z ) then we can write −→ v = ( x, 0 , 0) + (0 ,y, 0) + (0 , 0 ,z ) = x (1 , 0 , 0) + y (0 , 1 , 0) + z (0 , 0 , 1) . This means that any vector in R 3 can be expressed as a sum of scalar multiples (or linear combination ) of three speciFc vectors, known as the standard basis for R 3 , and denoted −→ ı = (1 , 0 , 0) −→ = (0 , 1 , 0) −→ k = (0 , 0 , 1) . It is easy to see that these are the vectors of length 1 pointing along the three (positive) coordinate axes (see ±igure 1.22 ). Thus, every vector x y z −→ ı −→ −→ k −→ v ( x ) −→ ı ( y ) −→ ( z ) −→ k ±igure 1.22: The Standard Basis for R 3 −→ v R 3 can be expressed as −→ v = x −→
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Unformatted text preview: + y + z k . We shall Fnd it convenient to move freely between the coordinate notation v = ( x,y,z ) and the arrow notation v = x + y + z k ; generally, we adopt coordinate notation when v is regarded as a position vector, and arrow notation when we want to picture it as an arrow in space. We began by thinking of a vector v in R 3 as determined by its magnitude and its direction, and have ended up thinking of it as a triple of numbers....
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