Engineering Calculus Notes 40

Engineering - whether two vectors point in parallel directions Given(nonzero 5 vectors −→ v and −→ w the respective unit vectors in the

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28 CHAPTER 1. COORDINATES AND VECTORS To come full circle, we recall that the vector −→ v = ( x,y,z ) has as its standard representation the arrow −−→ O P from the origin O to the point P with coordinates ( x,y,z ); thus its magnitude (or length , denoted v v -→ v v v ) is given by the distance formula | −→ v | = r x 2 + y 2 + z 2 . When we want to specify the direction of −→ v , we “ point ”, using as our standard representation the unit vector —that is, the vector of length 1—in the direction of −→ v . From the scaling property of multiplication by real numbers, we see that the unit vector in the direction of a vector −→ v ( −→ v n = −→ 0 ) is −→ u = 1 | −→ v | −→ v . In particular, the standard basis vectors −→ ı , −→ , and −→ k are unit vectors along the (positive) coordinate axes. This formula for unit vectors gives us an easy criterion for deciding
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Unformatted text preview: whether two vectors point in parallel directions. Given (nonzero 5 ) vectors −→ v and −→ w , the respective unit vectors in the same direction are −→ u −→ v = 1 | −→ v | −→ v −→ u −→ w = 1 | −→ w | −→ w. The two vectors −→ v and −→ w point in the same direction precisely if the two unit vectors are equal −→ u −→ v = −→ u −→ w = −→ u or −→ v = | −→ v | −→ u −→ w = | −→ w | −→ u . This can also be expressed as −→ v = λ −→ w −→ w = 1 λ −→ v 5 A vector is nonzero if it is not equal to the zero vector. Thus, some of its entries can be zero, but not all of them....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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