Question 91469
To calculate the angle between the vectors *[Tex \LARGE <4,-3>] and *[Tex \LARGE <4,3>] use this formula:


*[Tex \LARGE \cos(\theta)=\frac{\v{a}\cdot\v{b}}{\left|\v{a}\right|\left|\v{b}\right|}] where   *[Tex \Large \left|\v{a}\right|] is the magnitude (ie length) of vector *[Tex \Large \v{a}] and *[Tex \Large \left|\v{b}\right|] is the magnitude (ie length) of vector *[Tex \Large \v{b}] 


*[Tex \LARGE \cos(\theta)=\frac{4*4+-3*3}{sqrt{4*4+-3*-3}sqrt{4*4+3*3}}=\frac{16+-9}{sqrt{16+9}sqrt{16+9}}=\frac{7}{sqrt{25}sqrt{25}}=\frac{7}{sqrt{625}}]Calculate the dot product in the numerator and the magnitudes in the denominator



So we have this equation:


*[Tex \LARGE \cos(\theta)=\frac{7}{25}]


Now take the arccosine of both sides to isolate *[Tex \Large \theta]


*[Tex \LARGE \cos^{-1}\left(\cos(\theta)\right)=\cos^{-1}\left(\frac{7}{25}\right)]


*[Tex \LARGE \theta=\cos^{-1}\left(\frac{7}{25}\right)=\cos^{-1}\left(0.28\left)=1.28700221758657]


So


*[Tex \LARGE \theta=1.28700221758657] radians


The angle in degrees is:


*[Tex \LARGE \theta=(1.28700221758657)*\frac{180}{\pi}=73.739795291688]


So the angle is

*[Tex \LARGE \theta=73.739795291688] degrees