Answer:
The resultant of these two forces will be the greatest when the two forces are in the same direction.
Explanation:
Let [tex]F_1[/tex] and [tex]F_2[/tex] denote these two forces. The resultant of these two force would be the vector sum [tex](F_1 + F_2)[/tex].
Refer to the diagram attached. Join the head of [tex]F_1[/tex] to the tail of [tex]F_2[/tex]. Under this construction, the vector connecting the tail of [tex]F_1\![/tex] to the head of [tex]F_2\![/tex] (the vector shown with a lighter color) would represent the resultant vector [tex](F_1 + F_2)[/tex].
Consider the triangle inequality: as long as [tex]F_1[/tex] and [tex]F_2[/tex] (the first two sides of the triangle in the diagram) are not parallel to one another, the magnitude of the resultant vector would always be smaller than the sum of the magnitude of [tex]F_1\![/tex] and that of [tex]F_2\![/tex].
In other words: [tex]\| F_1 + F_2 \| < \| F_1 \| + \| F_2 \|[/tex].
However, notice that when [tex]F_1[/tex] and [tex]F_2[/tex] are in the same direction, the three sides [tex]F_1\![/tex], [tex]F_2\![/tex], and [tex](F_1 + F_2)[/tex] are in the same line and no longer form a triangle.
On the other hand, if [tex]F_1[/tex] and [tex]F_2[/tex] are opposite to one another, [tex]\| F_1 + F_2 \| = \| F_1 \| - \| F_2 \|[/tex].
The magnitudes of the two forces, [tex]\| F_1 \| = 5\; \rm N[/tex] and [tex]\| F_2 \|= 3\; \rm N[/tex], are fixed. Hence, when the two forces are in the same direction, [tex]\| F_1 + F_2 \| = \| F_1 \| + \| F_2 \| =8\; \rm N[/tex].
As long as [tex]F_1[/tex] and [tex]F_2[/tex] are not in the same line, the triangle inequality ensures that [tex]\| F_1 + F_2 \| < \| F_1 \| + \| F_2 \| =8\; \rm N[/tex].
Hence, [tex]8 \rm \; N[/tex] would be the greatest magnitude that the resultant force could achieve. That value is reached when the two forces are in the same direction.
