Question

Given v= 3i + 7j and w= – 4i - j, find the angle between v and w.

Answer 1

Given:

`\begin{gathered} v=3i+7j \\ w=-4i-j \end{gathered}`

To find the angle between v and w:

Using the formula,

`\begin{gathered} \cos \theta=(v\cdot w)/(|v\mleft\Vert w\mright|) \\ \cos \theta=((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*17}} \\ =\frac{-19}{\sqrt[]{986}} \end{gathered}`

Rationalising the denomiantor, we get

`\begin{gathered} \cos \theta=\frac{-19}{\sqrt[]{989}}*\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}`

Hence, the angle is,

`127.2348^(\circ)`