The resultant of two forces of magnitude \(2 \mathrm{~N}\) and \(3 \mathrm{~N}\) can never be. (A) \(4 \mathrm{~N}\) (B) \(1 \mathrm{~N}\) (C) \(2.5 \mathrm{~N}\) (D) \((1 / 2) \mathrm{N}\)

The triangle law of vector addition states that if two vectors (in this case, forces) are represented as two sides of a triangle taken in order, then the third side taken in reverse order represents the sum of the vectors (resultant force). In other words, if we draw the two force vectors as two sides of a triangle, then the magnitude and direction of the resultant force can be determined from the third side of the triangle.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side. This implies that for any two sides a and b, and the third side c, the following must hold true: \(a+b > c\) \(a-c < b\) \(b-c < a\) For our problem, we have two force vectors with magnitudes of 2 N and 3 N. Let R be the magnitude of the resultant force. Using the triangle inequality theorem, we can determine the possible range of values for the magnitude of the resultant force: \(2 + 3 > R\) \(2 - 3 < R\) \(3 - 2 < R\) From these inequalities, we can deduce that the magnitude of the resultant force R must lie within the following range: \(1 < R < 5\)

Now, let's check each of the given options to determine which one lies outside the possible range of magnitudes for the resultant force: (A) 4 N: This value is within the range (1, 5), so it could be the magnitude of the resultant force. (B) 1 N: This value is equal to the lower bound of the range but is not strictly greater than 1, meaning that it cannot be the magnitude of the resultant force. (C) 2.5 N: This value is within the range (1, 5), so it could be the magnitude of the resultant force. (D) (1/2) N: This value is less than the lower bound of the range (1, 5), meaning that it cannot be the magnitude of the resultant force. Thus, the correct answer is (B) 1 N, as it is the only option that satisfies the conditions for the resultant force magnitude.