Given:
[tex]\begin{gathered} v=3i+7j \\ w=-4i-j \end{gathered}[/tex]
To find the angle between v and w:
Using the formula,
[tex]\begin{gathered} \cos \theta=\frac{v\cdot w}{|v\mleft\Vert w\mright|} \\ \cos \theta=\frac{(3i+7j)\cdot(-4i-j)}{|3i+7j||-4i-j|} \\ \cos \theta=\frac{3(-4)+7(-1)}{\sqrt[]{3^2+7^2}\sqrt[]{(-4)^2+(-1)^2}} \\ =\frac{-12-7}{\sqrt[]{9+49}\sqrt[]{16+1}} \\ =\frac{-19}{\sqrt[]{58\times17}} \\ =\frac{-19}{\sqrt[]{986}} \end{gathered}[/tex]
Rationalising the denomiantor, we get
[tex]\begin{gathered} \cos \theta=\frac{-19}{\sqrt[]{989}}\times\frac{\sqrt[]{986}}{\sqrt[]{986}} \\ cos\theta=\frac{-19\sqrt[]{986}}{986} \\ \theta=\cos ^{-1}(\frac{-19\sqrt[]{986}}{986}) \\ \theta=127.2348 \end{gathered}[/tex]
Hence, the angle is,
[tex]127.2348^{\circ}[/tex]
