### Video Transcript

Resolve a force of 81 newtons into two perpendicular components, ๐น one and ๐น two, as shown in the figure. Give your answer correct to two decimal places.

In order to solve this problem, we can use Lamiโs theorem. This states that if three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

This means that if we have three forces ๐ด, ๐ต, and ๐ถ, then ๐ด divided by sin ๐ผ, the angle between force ๐ต and force ๐ถ, is equal to ๐ต divided by sin ๐ฝ, which is equal to ๐ถ divided by sin ๐พ.

Our first step in the figure shown is to work out the missing angle in the diagram. Well, if two forces are perpendicular to each other, then they must meet at 90 degrees. This means that the angle between the force ๐น two and the 81-newton force is 36 degrees, as 90 minus 54 is equal to 36.

Applying Lamiโs theorem gives us ๐น one divided by sin 36, as the 36-degree angle is between the force ๐น two and 81. This is equal to ๐น two divided by sin 54, as the 54-degree angle is between the force 81 newtons and ๐น one. This is also equal to 81 divided by sin 90, as the 90-degree angle is between the forces ๐น one and ๐น two.

To calculate the force ๐น one we will solve ๐น one divided by sin 36 is equal to 81 divided by sin 90. Multiplying both sides of this equation by sin 36 gives us that ๐น one is equal to 81 divided by sin 90 multiplied by sin 36. This gives us a value for the force ๐น one of 47.61 newtons to two decimal places.

In order to calculate the force ๐น two, we will solve the equation ๐น two divided by sin 54 is equal to 81 divided by sin 90. Multiplying both sides of this equation by sin 54 gives us ๐น two is equal to 81 divided by sin 90 multiplied by sin 54. This is equal to 65.32 newtons. Therefore, ๐น two is equal to 65.32 newtons to two decimal places.

The two perpendicular components ๐น one and ๐น two, shown in the figure, are 47.61 newtons and 65.32 newtons.