1. ## vector problem!

i apologise if this is the incorrect forum, i assume it would go under here or perhaps calculus forum - vector help!

find the formula for $\displaystyle ||b -a||^2$ in terms of $\displaystyle ||b||^2$, $\displaystyle ||a||^2$ and $\displaystyle b·a$

2. Originally Posted by iExcavate
i apologise if this is the incorrect forum, i assume it would go under here or perhaps calculus forum - vector help!

find the formula for $\displaystyle ||b -a||^2$ in terms of $\displaystyle ||b||^2$, $\displaystyle ||a||^2$ and $\displaystyle b·a$

This question belongs, me believes ,in trigonometry-analytic geometry. Anyway:

$\displaystyle b-a$ is an arrow beginning at $\displaystyle a$ and ending at $\displaystyle b$ and $\displaystyle \|b-a\|$ is its length, so using the cosines law (theorem) on the resulting triangle we get:

$\displaystyle \|b-a\|^2=\|a\|^2+\|b\|^2-2\|a\|\|b\|\cos \theta$

I'm not sure what is $\displaystyle ba$, but if you meant the dot product of these two vectors then just remember that $\displaystyle \cos \theta=\frac{b\cdot a}{\|b\|\|a\|}$ , so you can substitute above.

Tonio

3. how do i deduce $\displaystyle \|b-a\|^2 = \|b\|^2 + \|a\|^2 - 2\|a\|\|b\|cos\theta$
to prove:$\displaystyle \cos \theta=\frac{b\cdot a}{\|b\|\|a\|}$

4. tonio told you: apply the cosine law to the triangle with sides a, b, and a- b.