Answer :
Answer:
The angle between the given vectors u and v is [tex]\theta=cos^{-1}\left[\frac{3}{\sqrt{10}}\right][/tex]
Step-by-step explanation:
Given vectors are [tex]\overrightarrow{u}=(3,4)[/tex] and [tex]\overrightarrow{v}=(1,3)[/tex]
Now compute the dot product of u and v:
[tex]\overrightarrow{u}.\overrightarrow{v}=(3,4).(1,3)[/tex]
[tex]=(3)(1)+(4)(3)[/tex]
[tex]=3+12[/tex]
[tex]=15[/tex]
Now find the magnitude of u and v:
[tex]|\overrightarrow{u}|=\sqrt{3^2+4^2}[/tex]
[tex]=\sqrt{9+16}[/tex]
[tex]=\sqrt{25}[/tex]
[tex]=5[/tex]
[tex]|\overrightarrow{u}|=5[/tex]
[tex]|\overrightarrow{v}|=\sqrt{1^2+3^2}[/tex]
[tex]=\sqrt{1+9}[/tex]
[tex]=\sqrt{10}[/tex]
[tex]|\overrightarrow{v}|=\sqrt{10}[/tex]
To find the angle between the given vectors
[tex]\overrightarrow{u}.\overrightarrow{v}=|\overrightarrow{u}|\overrightarrow{v}|cos\theta[/tex]
[tex]\theta=cos^{-1}\left[\frac{\overrightarrow{u}.\overrightarrow{v}}{|\overrightarrow{u}|\overrightarrow{v}|}\right][/tex]
[tex]=cos^{-1}\left[\frac{15}{5\times \sqrt{10}}\right][/tex]
[tex]=cos^{-1}\left[\frac{15}{5\times \sqrt{10}}\right][/tex]
[tex]\theta=cos^{-1}\left[\frac{3}{\sqrt{10}}\right][/tex]
Therefore the angle between the vectors u and v is
[tex]\theta=cos^{-1}\left[\frac{3}{\sqrt{10}}\right][/tex]