## To calculate the Cartesian components of the forces F[1] and F[2], you need to use the given magnitudes and directions.

For F[1], the magnitude is 3âˆš5 newtons and the direction is parallel to the vector i + 2j. To calculate the Cartesian components, you first need to find the unit vector in the direction of i + 2j. To do this, you divide each component of the vector by its magnitude:

u[1] = (i + 2j) / âˆš(1^2 + 2^2) = (i + 2j) / âˆš(5)

Then, you multiply the unit vector by the magnitude of the force:

F[1] = (3âˆš5) * u[1] = (3âˆš5) * [(i + 2j) / âˆš(5)] = 3i + 6j

So, the Cartesian components of F[1] are 3i + 6j.

Similarly, for F[2], the magnitude is âˆš5 newtons and the direction is parallel to the vector i - 2j. Following the same steps as above, we can calculate the Cartesian components of F[2]:

u[2] = (i - 2j) / âˆš(1^2 + (-2)^2) = (i - 2j) / âˆš(5)

F[2] = (âˆš5) * u[2] = (âˆš5) * [(i - 2j) / âˆš5] = i - 2j

So, the Cartesian components of F[2] are i - 2j.

To calculate the total force F[1] + F[2], simply add the Cartesian components together:

F[1] + F[2] = (3i + 6j) + (i - 2j) = 4i + 4j

Therefore, the total force acting on the particle in component form is 4i + 4j.

Regarding the part about showing the couple of the total force about the point with position vector i is zero, a couple consists of two equal and opposite forces acting at different points and parallel to each other. In this case, since only one total force is given, it cannot form a couple. Therefore, the couple about the point with position vector i is indeed zero.