### Video Transcript

The forces ๐น one, which is equal to two ๐ plus seven ๐; ๐น two, which is equal to ๐๐ minus six ๐; and ๐น three, which is equal to six ๐ plus ๐ plus eight ๐, act on a particle, where ๐ and ๐ are two perpendicular unit vectors. Given that the system is in equilibrium, determine the values of ๐ and ๐.

If the system is in equilibrium, then the resultant force must equal zero. This means that the ๐-components โ two ๐, ๐๐, and six ๐ โ must equal zero. The coefficients are two, ๐, and six. Therefore, two plus ๐ plus six equals zero. Two plus six is equal to eight. Therefore, ๐ plus eight is equal to zero. Subtracting eight from both sides of the equation gives us a value of ๐ of negative eight. This means that the force ๐น two is negative eight ๐ minus six ๐.

As the ๐-components must also equal zero, seven ๐ minus six ๐ and ๐ plus eight ๐ must equal zero. Once again, the coefficients are seven, negative six, and ๐ plus eight. Seven minus six plus ๐ plus eight equals zero. Seven take away six is one. One plus eight is equal to nine. Therefore, ๐ plus nine equals zero. Subtracting nine from both sides of this equation gives us a value of ๐ equal to negative nine.

This means that the force ๐น three is equal to six ๐ plus negative nine plus eight ๐. Negative nine plus eight is negative one. Therefore, ๐น three is six ๐ minus one ๐. If the three forces ๐น one, ๐น two, and ๐น three are acting on a particle where the system is in equilibrium, then the value of ๐ is negative eight and the value of ๐ is negative nine.