Final answer:
The vector added to A to get R as a resultant is such that it combines vectorially with A to produce R. In case both forces are in the same or opposite directions on a line, the resultant is their sum or the absolute value of their difference, and direction is determined with the chosen positive direction. The head-to-tail method is employed for such vector addition.
Step-by-step explanation:
The vector added to A to give R as a resultant must be such that when it is combined with A, it produces R. For two forces, let's say A and a second force B, to combine to form a resultant force R, the forces must be added vectorially. In a one-dimensional example, if A and B are in the same direction, their resultant R would be the sum of the magnitudes of A and B. However, if they are in opposite directions, one force is considered positive and the other negative, depending on the chosen positive direction.
When working with vectors in one dimension or two, we can use the head-to-tail method of vector addition to obtain the resultant vector R. This method is both commutative and associative, meaning that A + B = B + A and the order in which multiple vectors are added does not change the resultant. Using graphical representation, the vector B would be drawn such that its tail starts at the head of vector A, and the resultant R is drawn from the tail of A to the head of B. The resulting magnitude and direction are then found using tools like a ruler and a protractor.
If vectors A and B are parallel, the magnitude of the resultant R will be the sum of their magnitudes. If they are antiparallel (opposite directions but along the same line), the magnitude of the difference will be the absolute value of the difference between the magnitudes of A and B, and the direction of the resultant vector is in the direction of the longer vector.