What is the mistake in my derivation of cosine

Question

In a triangle $ABC$ let's let $\overrightarrow{a}=\text{vector }\overrightarrow{BC}$, let $\overrightarrow{b}=\text{vector }\overrightarrow{CA}$, and let $\overrightarrow{c}=\text{vector }\overrightarrow{BA}$,. Let's let $\gamma=\text{angle }ACB$.

Then;

$$\overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b}$$

$$\overrightarrow{c}\cdot\overrightarrow{c}=\left(\overrightarrow{a}+\overrightarrow{b}\right)\cdot\left(\overrightarrow{a}+\overrightarrow{b}\right)$$

$$\overrightarrow{c}\cdot\overrightarrow{c}=\overrightarrow{a}\cdot\overrightarrow{a}+\overrightarrow{b}\cdot\overrightarrow{b}+2\overrightarrow{a}\cdot\overrightarrow{b}$$

$$|c|^2=|a|^2+|b|^2+2|a||b|\cos(\gamma)$$

But, the cosine law says: $|c|^2=|a|^2+|b|^2-2|a||b|\cos(\gamma)$.

So I guess I made a mistake somewhere in the process. What was my mistake?

Answer 1

Because the angle ACB should be the angle formed by vector CA and CB instead of vector CA and vector BC, that was where you made the mistake.

Answer 2

If $\gamma$ is defined to be angle $ACB$, then $\gamma$ is not the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. The angle between them will be $180^{\circ}-\gamma$.