A, B, C construct an equilateral triangle.Check if the statements are true or not.

Question

Let A, B, C construct an equilateral triangle. I want to check if the following statements are true or not.

1. $\vec{AB}+\vec{AC}=\vec{BC}$

   I think this is wrong because: $\vec{AB}=\vec{AC}+\vec{CB}=\vec{AC}-\vec{BC}\Rightarrow -\vec{AB}+\vec{AC}=\vec{BC}$.

2. $2\vec{AB}+\vec{AC}=\vec{AC}+\vec{BC}$

   That would mean that $2\vec{AB}=\vec{BC}$, or not? I think that this is wrong since I don't see how this could be true. So, I think that this statement is wrong.

3. $\vec{AB}\cdot \vec{BC}=\vec{AC}\cdot \vec{CB}$

   This is correct since the dot product is equal to the product of the length of the two vectors and the angle, which is the same at both sides of equality.

4. $|\vec{AB}+\vec{AC}|=2|\vec{BC}|$

   Could you give me a hint for this one?

5. $\vec{AB}-\vec{AC}=\vec{BC}$

   This is wrong, since it should be $\vec{AB}-\vec{AC}=-\vec{BA}-\vec{AC}=-(\vec{BA}+\vec{AC})=-\vec{BC}=\vec{CB}$.

$$$$ Are my thoughts correct?

Answer 1

In part 4, $|\vec{AB}+\vec{AC}|=a\sqrt{3}$ were $a$ is the length of the side of the equileteral triangle (You can check that by applying Pythagoras theorem in the semiequiletral triangle that is formed by drawing a height (median) in the triangle $ABC$)

So you'll get that $|\vec{AB}+\vec{AC}|=a\sqrt{3}$ And $2|\vec{BC}|=2a$

So as a result we get that $|\vec{AB}+\vec{AC}| \lt 2|\vec{BC}|$

Except for $a=0$

And the other 4 ideas are correct.