I have encountered such problem that ask us to calculate the angle between two vectors:
$\vec{u},\vec{v},\vec{w}$ has a length of $5$,$12$,$15$ respectively. Their sum gives a $\vec{0}$. What is the angle between $\vec{u}$ and $ \vec{v}$ ?
My first thought is to use cosine law, which will be as following:
- $||\vec{w}||^2 = ||\vec{u}||^2+||\vec{v}||^2 - 2||\vec{u}|| ||\vec{v}||\cos(\theta)$
- $\theta = \frac{||\vec{w}||^2 - ||\vec{u}||^2-||\vec{v}||^2}{- 2||\vec{u}|| ||\vec{v}||} $
- $\theta = \arccos(\frac{-7}{15}) $
However, the answer that the books give is $\arccos(\frac{7}{15})$.
Also, this is the procedure provided by the book, which I found very confusing and I couldn't understand (For exemple: Why did they come up with first two setps? What do they even mean?)
And in the procedure provided, I found the book eventually use almost the same formula but with $2||\vec{u}|| ||\vec{v}||\cos(\theta)$ instead of $- 2||\vec{u}|| ||\vec{v}||\cos(\theta)$
So I truly don't know where is my mistake, and why the book and I just have a difference of sign?
Really need help on this one! Thank you!
Here's why you get the supplementary angle.