Problem 1B-02: Position vectors, geometric approach

Use summation and subtraction of position vectors of the first kind as defined in Topics 1-7 to perform the following operations.

(a) Position vectors A and B are given in the following picture. Draw A+B.

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(b) Position Vectors C and D are given in the following picture. Draw C-D.

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(c) Position vectors A, B and C are given in the picture. Draw A+B-C. Does A-C+B has any physical meaning?

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Solution:

The solutions here are based on the summation and subtraction of position vectors of the first kind as defined in Topic 1-7.

(a) The answerer is shown as red position vector that originates from the origin and specify point P in the following picture.

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The procedure is to draw a parallelogram with A and B as two adjacent sides. Other two sides of the parallelogram are shown as blue lines in the above picture. The diagonal OP then is the desired answer A+B.

(b)

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Form a parallelogram with D as one side and C as a diagonal as shown in the above picture. The the other side originating from the origin 0 and ending at point P is the desired answer C-D.

(c)

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In the above picture first draw a parallelogram with A and B as two adjacent sides; other two sides of the parallelogram are shown as blue lines. The diagonal of the parallelogram originating from the origin is the sum A+B. Then form a parallelogram with C as one side and A+B as a diagonal; the other two sides of the parallelogram are colored in orange. The remaining side, colored in red is the answer A+B-C. To form the vector A-C+B we use the following picture.
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We first draw a parallelogram with A as a diagonal and C as one side; other two sides are colored orange. The fourth side colored blue is the position vector A-C. The parallelogram with A-C and B as two sides are drawn with other two sides colored blue. The diagonal of this last parallelogram, colored red, is the position vector A-C+B. Please note that the result of A-C+B constructed here is the same as A+B-C that is constructed in conjunction with the previous graph.