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.

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

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(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.

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(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.

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(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.

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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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