Question

Two forces are acting on an object at the same point. Determine the angle between the two forces. F1= <-2,-7> F2= <3,1>

Answer 1

Given:

The two forces are:

F1= <-2 , -7>

F2=< 3 , 1>

The angle between the two forces is given by the dot product of the two forces.

Thus,

`a\cdot b=\parallel a\parallel.\parallel b\parallel.\cos \theta`

Where, a and b are the two forces and theta is the angle between them

Substituting the values in this formula,

`\cos \theta=(F1\cdot F2)/(\parallel F1\parallel\cdot\parallel F2\parallel)`

Now,

`\begin{gathered} \parallel F1\parallel=\sqrt[]{(-2)^2+(-7)^2} \\ =\sqrt[]{4+49} \\ =\sqrt[]{53} \end{gathered}`
`\begin{gathered} \parallel F2\parallel=\sqrt[]{(3)^2+(1)^2} \\ =\sqrt[]{9+1} \\ =\sqrt[]{10} \end{gathered}`
`\begin{gathered} F1\cdot F2=-2*3+\text{-7}*1 \\ =-6-7 \\ =-13 \end{gathered}`

Hence, the angle between the forces is:

`\begin{gathered} \cos \theta=\frac{-13}{\sqrt[]{53}*\sqrt[]{10}} \\ =\frac{-13}{\sqrt[]{530}} \\ =(-13)/(23.021) \\ =-0.564 \end{gathered}`

Now,solving further:

`\begin{gathered} \theta=\cos ^(-1)(-0.564) \\ =124.332878^(\circ)+360k\text{ or 2.170018 rad+2}\pi k;k=\pm1,\pm2\ldots \end{gathered}`

Hence, the angle between the forces is 124.332878 degrees or 2.170018 radians.