Respuesta :
You have the following vectors:
[tex]\begin{gathered} v=-12x+5y \\ w=3x+10y \end{gathered}[/tex]By definition, the Dot product of two vectors
[tex]\begin{gathered} a=hx+ry \\ b=mx+py \\ \end{gathered}[/tex]is the following:
[tex]a\cdot b=h\cdot m+r\cdot p[/tex]Then you can calculate the Dot product with the vectors given in the exercise:
[tex]\begin{gathered} v\cdot w=(-12)(3)+(5)(10) \\ v\cdot w=-36+50 \\ v\cdot w=14 \end{gathered}[/tex]The Dot product between two vectors can be also written as:
[tex]v\cdot w=|v|\cdot|w|\cdot\cos \alpha[/tex]Where α is the angle between the vectors.
Now, in order to calculate the angle, you need to solve for the angle α:
[tex]\begin{gathered} \cos \alpha=\frac{v\cdot w}{|v|\cdot|w|} \\ \\ \alpha=\cos ^{-1}(\frac{v\cdot w}{|v|\cdot|w|}) \end{gathered}[/tex]You know that
[tex]v\cdot w=14[/tex]So now you need to find |v| amd |w|.
By definition:
[tex]\begin{gathered} |v|=\sqrt[]{(-12)^2+(5)^2}=13 \\ \\ |w|=\sqrt[]{(3)^2+(10)^2}=10.44 \end{gathered}[/tex]Knowing these values, you can calculate the angle:
[tex]\begin{gathered} \alpha=\cos ^{-1}(\frac{v\cdot w}{|v|\cdot|w|}) \\ \\ \alpha=\cos ^{-1}(\frac{14}{13\cdot10.44}) \\ \\ \alpha\approx84.1\degree \end{gathered}[/tex]The answer is:
[tex]84.1\degree[/tex]