Question

An object is acted upon by the forces F1=<10​,6​,4> and F2=<0​,3​,9>. Find the force F3 that must act on the object so that the sum of the forces is zero.

Answer 1

$\vec F_3=\langle x,y,z\rangle$ is a force vector such that

$\vec F_1+\vec F_2+\vec F_3=\vec0$

$\implies\langle10,6,4\rangle+\langle0,3,9\rangle+\langle x,y,z\rangle=\langle0,0,0\rangle$

$\implies\begin{cases}10+x=0\\9+y=0\\13+z=0\end{cases}\implies x=-10,y=-9,z=-13$

So the missing force is

$\boxed{\vec F_3=\langle-10,-9,-13\rangle}$