Prove 3 vectors are collinear

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If $A=(2,4), \; B=(8,6), \; C=(11,7)$, then $$ \vec{AB} = B - A = (6,2) \quad ; \quad \vec{BC} = C - B = (3,1) $$ So $\vec{AB} = 2 \vec{BC}$, which is different from your conclusion.

Anyway, this says that $\vec{AB}$ and $\vec{BC}$ are parallel, since one is just a multiple of the other. So, to get from $B$ to $C$, you go in the same direction you went to get from $A$ to $B$ (no turn is required). This means that $A$, $B$, $C$ must all lie on the same line.

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dagda1
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dagda1

Updated on August 01, 2022

Comments

  • dagda1
    dagda1 over 1 year

    I am asked to prove A(2,4), B(8,6), C(11,7) are collinear using vectors.

    I can work AB by subtracting A from B and BC by subtracting B from C in vector form.

    I can say that BC = 2AB.

    But I don't understand why this proves they are collinear.

    • bubba
      bubba over 7 years
      Your conclusion $BC =2 AB$ is not correct. Please see my answer.
    • bubba
      bubba over 7 years
      Also, your title is not really right. You're not proving three vectors, are collinear, you're proving three points are collinear.
  • bubba
    bubba over 7 years
    Since the OP is doing elementary vector arithmetic, he (or she) probably doesn't know what "affine space" or "affine dimension" mean.
  • goblin GONE
    goblin GONE over 7 years
    @bubba, well I wanted to explain the underlying idea, which is that you can answer questions about affine geometry by first subtracting by an appropriate vector so that you're now dealing with vector geometry. If this is all going over the OP's head, he/she can always just rely on your answer which "cuts straight to the chase" so to speak. But we live in the information age, and as always my hope is that the OP will Google unfamiliar terms and struggle with them until they're understood.