Problem 1B-3: Position vectors, the second kind of summation and subtraction

Follow the summation and the subtraction of the second kind for position vectors, as defined in Topic 1-7.

(a) Draw A+B in the following picture.


(b) Draw A-B in the following picture.


(c) Draw A+B-C in the following picture. How about A+B+C and A-C+B?







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Solution:

The definition of the second kind for the summation and the subtraction of position vectors as discussed in Topic 1-7 is used throughout this solution.

(a) According to the definition of the summation of position vectors of the second kind, the sum of A+B is drawn as a red arrow in the following picture. It is a position vector originating from the origin and specifying the position of point P.


(b) Following the definition of the subtraction of position vectors of the second kind as discussed in Topic 1-7, the subtraction A-B is plotted as a red arrow in the following picture. It is a position vector originating from point R and specifying the position of point P.


(c) According to the definition of the summation of the second kind, first the sum A+B is plotted in the following picture as a blue arrow; it is a position vector that originates from the origin and ends at point P. Then, using the definition of the subtraction of the second kind, the subtraction, (A+B)-C = A+B-C, is plotted as the red arrow; it is a position vector that originates from point R and ends at point P.


The vector A+B+C has no meaning here since the sum of the second kind between A+B and C is not defined.

The vector A-C is first drawn as a blue arrow that originates from point Q and ends at point R. Then according to the definition of the summation of the second kind, the vector A-C+B = (A-C)+Bis drawn as a red arrow, the position vector that originates from point Q and ends at point P. The result is the same as the case A+B-C.


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