How Do You Determine Scalars a and b from Vectors A, B, and C?

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To determine the scalars a and b from the vectors A, B, and C, the equation aA + bB + C = 0 is used, leading to aA + bB = -C. The components of the vectors are expressed, resulting in the equations 6a - 8b = 26 and 8a + 3b = 19. The discussion emphasizes the importance of equating vector components to solve for the unknowns. Participants clarify misunderstandings about vector magnitudes and the correct approach to solving the equations. The focus remains on solving the two equations to find the values of a and b.
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I saw this question posted yesterday, and now got a similar question to work out.

A = (6i-8j) cm
B = (-8i+3j) cm
C = (26i+19j) cm

aA+bB+C=0

Determine the two scalars a and b.

Ideas anyone??

Thanks

Dora
 
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C=0 ?? But you just said C=26i+19j . Is this a typo? or did you mean aA+bB-C=0

in which case aA+bB=C
seems pretty straightforward to me. Split it up into the vector components, and youll have 2 equations with 2 unknowns, easily solveable.
 
Sorry. That was a tipo. I made a mistake.

aA+bB+C=0 not aA+bB=C=0 not
 
Well, what have you done so far? How have you approached it?
 
I used the equation a^2 + b^2 = c^2 and the coodinates (6,-8) and (-8,3) to determine that the magnitude of A is 0.5cm and that the magnitude of B is 0.7cm. But I don't know if that is what is meant by "determine the two scalars a and b". I'm asuming scalars in this question is the scalar quantity or "magnitude".
 
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The "equation a^2+ b^2= c^2" doesn't even make sense here. You are given vectors A, B, C, not numbers a, b, c (and you certainly don't have any number c).

Do you know how to add vectors and multiply vectors by a number? That should have been ther first thing you learned!

If A= 6i+8j, then aA= (6a)i+ (8a)j.

If B= -8i+ 3j, then bB= (-8b)i+ (3b)j

aA+ bB = (6a- 8b)i+ (8a+ 3b)j and that must be equal to C= 26i+ 19j.

Okay, have you learned that two vectors are equal only if the respective components are equal?

To have aA+ bB= C, you must have (6a- 8b)i+ (8a+ 3b)j= 26i+ 19j and so
6a- 8b= 26 and -8a+ 3b= 19.

Can you solve those two equations for a and b?
 
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