Vectors

[rsgunter21]

Find the angle between the vectors a = <3,-2> and b = <-1,3>

Thanks!

Ryan

 

[soroban]

Hello, Ryan!

There is a formula for this problem . . .

Find the angle between the vectors \(\displaystyle \vec{a}\,=\,\langle3,-2\rangle\) and \(\displaystyle \vec{b}\,=\,\langle-1,3\rangle\)

The angle between two vectors \(\displaystyle \vec{u}\) and \(\displaystyle \vec{v}\) is given by: \(\displaystyle \L\;\cos\theta\;=\;\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}\)

We have: \(\displaystyle \,\vec{a}\cdot\vec{b}\;=\;\langle3,-2\rangle\cdot\langle-1,3\rangle\;=\;-3\,-\,6\;=\;-9\)

\(\displaystyle \;\;\)and: \(\displaystyle |\vec{a}|\;=\;\sqrt{3^2+(-2)^2}\;=\;\sqrt{13},\;\;|\vec{b}|\;=\;\sqrt{(-1)^2+3^2}\;=\;\sqrt{10}\)

So we have: \(\displaystyle \,\cos\theta\;=\;\frac{-9}{\sqrt{130}}\)

Therefore: \(\displaystyle \,\theta\;=\;\cos^{-1}\left(\frac{-9}{\sqrt{130}}\right)\;\approx\;142.1^o\)

 

[rsgunter21]

Awesome, thanks.