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,

$\cos \theta=\frac{F1\cdot F2}{\parallel F1\parallel\cdot\parallel F2\parallel}$

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\times3+\text{-7}\times1 \\ =-6-7 \\ =-13 \end{gathered}$

Hence, the angle between the forces is:

$\begin{gathered} \cos \theta=\frac{-13}{\sqrt[]{53}\times\sqrt[]{10}} \\ =\frac{-13}{\sqrt[]{530}} \\ =\frac{-13}{23.021} \\ =-0.564 \end{gathered}$

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.