Answer :
Answer:
The angle between the vectors is [tex]\theta=\cos^{-1}\frac{-3}{\sqrt{34}}[/tex]
Step-by-step explanation:
Given : Vectors [tex]u=(0,4)[/tex] and [tex]v=(5,-3)[/tex]
To find : The angle between the vectors?
Solution :
The formula to found the angle between the vector is
[tex]\cos\theta=\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}[/tex]
[tex]u=(0,4)[/tex]
[tex]\vec{u}=0i+4j[/tex]
[tex]v=(5,-3)[/tex]
[tex]\vec{v}=5i-3j[/tex]
[tex]\vec{u}\cdot \vec{v}=0\times 5+4\times -3[/tex]
[tex]\vec{u}\cdot \vec{v}=0-12[/tex]
[tex]\vec{u}\cdot \vec{v}=-12[/tex]
[tex]|\vec{u}|=\sqrt{0^2+4^2}=\sqrt{16}=4[/tex]
[tex]|\vec{v}|=\sqrt{5^2+(-3)^2}=\sqrt{25+9}=\sqrt{34}[/tex]
Substitute in the formula,
[tex]\cos\theta=\frac{-12}{4\sqrt{34}}[/tex]
[tex]\cos\theta=\frac{-3}{\sqrt{34}}[/tex]
[tex]\theta=\cos^{-1}\frac{-3}{\sqrt{34}}[/tex]
Therefore, The angle between the vectors is [tex]\theta=\cos^{-1}\frac{-3}{\sqrt{34}}[/tex]