The Position Vector. Using the Cartesian coordinate system, the position vector can be explicitly written as:

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Ethelbert Austin
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1 8/23/2005 The Position Vecto.doc 1/7 The Position Vecto Conside a point whose location in space is specified with Catesian coodinates (e.g., P(x,y,)). Now conside the diected distance (a vecto quantity!) extending fom the oigin to this point. P(x,y,) y x This paticula diected distance a vecto beginning at the oigin and extending outwad to a point is a vey impotant and fundamental diected distance known as the position vecto. Using the Catesian coodinate system, the position vecto can be explicitly witten as: x ˆ a + y ˆ a + ˆ a =

2 8/23/2005 The Position Vecto.doc 2/7 * Note that given the coodinates of some point (e.g., x =1, y =2, =-3), we can easily detemine the coesponding position vecto (e.g., = â + 2â 3â ). * Moeove, given some specific position vecto (e.g., = 4â 2â ), we can easily detemine the coesponding y coodinates of that point (e.g., x =0, y =4, =-2). In othe wods, a position vecto is an altenative way to denote the location of a point in space! We can use thee coodinate values to specify a point s location, o we can use a single position vecto. I see! The position vecto is essentially a pointe. Look at the end of the vecto, and you will find the point specified! P( )

3 8/23/2005 The Position Vecto.doc 3/7 The magnitude of Note the magnitude of any and all position vectos is: = = x + y + = The magnitude of the position vecto is equal to the coodinate value of the point the position vecto is pointing to! Q: Hey, this makes pefect sense! Doesn t the coodinate value have a physical intepetation as the distance between the point and the oigin? A: That s ight! The magnitude of a diected distance vecto is equal to the distance between the two points in this case the distance between the specified point and the oigin! Altenative foms of the position vecto Be caeful! Although the position vecto is coectly expessed as: = xaˆ ˆ ˆ x + yay + a

4 8/23/2005 The Position Vecto.doc 4/7 It is NOT CORRECT to expess the position vecto as: no ρ ˆ a + φ ˆ a + ˆ a p φ aˆ + θ ˆ a + φ ˆ a θ φ NEVER, EVER expess the position vecto in eithe of these two ways! It should be eadily appaent that the two expession above cannot epesent a position vecto because neithe is even a diected distance! Q: Why sue it is of couse eadily appaent to me but why don t you go ahead and explain it to those with less insight! A: Recall that the magnitude of the position vecto has units of distance. Thus, the scala components of the position vecto must also have units of distance (e.g., metes). The coodinates x, y,, ρ and do have units of distance, but coodinates θ and φ do not. Thus, the vectos θ a ˆθ and φ a ˆφ cannot be vecto components of a position vecto o fo that matte, any othe diected distance!

5 8/23/2005 The Position Vecto.doc 5/7 Instead, we can use coodinate tansfoms to show that: = x ˆ a + y ˆ a + ˆ a = ρcosφ ˆ a + ρsinφ ˆ a + ˆ a = sinθ cosφ ˆ a + sinθ sinφ ˆ a + cosθ ˆ a ALWAYS use one of these thee expessions of a position vecto!! Note that in each of the thee expessions above, we use Catesian base vectos. The scala components can be expessed using Catesian, cylindical, o spheical coodinates, but we must always use Catesian base vectos. Q: Why must we always use Catesian base vectos? You said that we could expess any vecto using spheical o base vectos. Doesn t this also apply to position vectos? A: The eason we only use Catesian base vectos fo constucting a position vecto is that Catesian base vectos ae the only base vectos whose diections ae fixed independent of position in space!

6 8/23/2005 The Position Vecto.doc 6/7 To see why this is impotant, let s go ahead and change the base vectos used to expess the position vecto fom Catesian to spheical o cylindical. If we do this, we find: = x ˆ a + y ˆ a + ˆ a = ρ ˆ a + ˆ a = ˆ a ρ Thus, the position vecto expessed with the cylindical coodinate system is = ρ ˆ a + ˆ a, while with the spheical coodinate system we get ρ ˆ = a. The poblem with these two expessions is that the diection of base vectos a ˆρ and a ˆ ae not constant. Instead, they themselves ae vecto fields thei diection is a function of position! Thus, an expession such as = 6 ˆ a does not explicitly define a point in space, as we do not know in what diection base vecto a ˆ is pointing! The expession = 6 ˆ a does tell us that the coodinate =6, but how do we detemine what the values of coodinates θ o φ ae? (answe: we can t! ) Compae this to the expession: = ˆ a + 2 ˆ a 3 ˆ a

7 8/23/2005 The Position Vecto.doc 7/7 Hee, the point descibed by the position vecto is clea and unambiguous. This position vecto identifies the point P(x =1, y =2, =-3). Lesson leaned: Always expess a position vecto using Catesian base vectos (see box on pevious page)!

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita