13 The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are acting on a particle. $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 25 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }
& \mathbf { F } _ { 2 } = ( - 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { N }
& \mathbf { F } _ { 3 } = ( 15 \mathbf { i } - 28 \mathbf { j } ) \mathrm { N } \end{aligned}$$ The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.
The resultant of these three forces is \(\mathbf { F }\) newtons. 13

1. 1. Find the magnitude of F, giving your answer to three significant figures.
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2. (ii) Find the acute angle that \(\mathbf { F }\) makes with the horizontal, giving your answer to the nearest \(0.1 ^ { \circ }\)
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3. The fourth force, \(F _ { 4 }\), is applied to the particle so that the four forces are in equilibrium. Find \(\mathbf { F } _ { 4 }\), giving your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
   [0pt] [1 mark]
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