When can we say that the resultant of the two vectors is maximum

The correct option is: Both the vectors are acting in the same direction

Let us understand why this is the correct answer by looking at the mathematical formula for the resultant of two vectors.

Consider two vectors, $A$ and $B$, with an angle $\theta$ between them. According to the vector addition law, the magnitude of their resultant vector $R$ is given by the formula:

$R=\sqrt{A^2+B^2+2AB\cos \left(\theta \right)}$

In this formula, the magnitudes of the vectors $A$ and $B$ are constant, positive values. Therefore, the magnitude of the resultant $R$ depends entirely on the value of $\cos \left(\theta \right)$.

To make the resultant $R$ maximum, the value of $\cos \left(\theta \right)$ must be at its maximum possible value. The maximum value of the cosine function is $1$, which occurs when the angle $\theta =0°$.

An angle of $\theta =0°$ means that the two vectors are pointing in the exact same direction. Substituting $\cos (0°)=1$ back into the resultant formula yields:

$R_{\max }=\sqrt{A^2+B^2+2AB\left(1\right)}$

$R_{\max }=\sqrt{{(A+B)}^2}$

$R_{\max }=A+B$

Thus, when both vectors act in the same direction, their magnitudes add up directly, resulting in the maximum possible resultant magnitude.