Two forces F₁ and F₂ act on a particle P - Edexcel - A-Level Maths Mechanics - Question 7 - 2016 - Paper 1

To find the force F₂, we need the resultant force F₁ + F₂ to act in the direction of (i + 3j).

1. Express F₁ and F₂:

   • We have F₁ = (-1i + 2j) N.
   • Let F₂ = (xi + yj), where x and y are to be determined.

2. Set up the equation for the resultant:

   • The resultant R = F₁ + F₂ = (-1 + x)i + (2 + y)j.
   • Given that the resultant is in the direction of (i + 3j), we can express this condition mathematically:
   • R = k(1i + 3j) for some scalar k.

3. Equate the components:

   • From the i-component: $-1 + x = k$
   • From the j-component: $2 + y = 3k$

4. Express k:

   • From $-1 + x = k$, we have $k = x - 1$.
   • Substitute for k in the j-component: $2 + y = 3(x - 1)$.

5. Solve for y in terms of x:

   • Rearranging gives us, $y = 3x - 5$.

6. Find specific components by substituting:

   • The resultant must be consistent with both equations from the resultant vector. Hence:
     1. Substitute y back into the equations, leading to system of equations;
     2. Solve for k = 2, thus setting: $-1 + x = 2$ $x = 3$ and for y, from $y = 3(3) - 5$, we find: $y = 4$.

7. Final result:

   • Thus, the resultant force F₂ is: $F₂ = (3i + 4j) N.$