Vectors A, B, and C satisfy the equation A + B = C, and their magnitudes are related by the equation A + B = C. How is the vector A oriented with respect to vector B ? Give reason for your answer.
Answers
Vectors A and B are parallel and/or colinears (they lay on the same line or parallel lines) and have the same orientation.
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Graphically, the sum of two non-colinear vectors A and B would form a triangle, with sides of length A, B and C (scalar values). There is a theorem from Geometry that says (not exactly with this words, but it still works):
“In any triangle, the length of one side is ALWAYS less than the sum of the other two.”
So, if this is true, is IMPOSSIBLE (in R^n, at least) for the scalar equation A + B = C to be true. In fact, A + B > C.
Then, considering A and B to be colinears, it can happen the following:
They have the same orientation. In this case, the sum of the two vectors would result in a vector with the same orientation, and a length that is the sum of the lengths of A and B. So, the conditions are satisfied.
They have opposite orientation. In this case, the equation A + B = C is transformed into A - B = C, and the condition is not satisfied.
P.S.: Sorry for my broken English.