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Vector addition is the process of combining two or more vectors to find a resultant vector. This operation is crucial in physics, where vectors represent quantities like velocity, force, or displacement that have both magnitude and direction.

To add vectors graphically, they are usually arranged tip-to-tail, which means the tail of one vector is positioned at the tip of the previous one. For example, if one was to walk 5 meters east and then 5 meters north, the resultant displacement would be found by drawing a vector east, then drawing a vector north from where the first vector ends. The diagonal from the start to the end point is the resultant vector, illustrating the total displacement.

In the case of adding velocities, if two objects moving at velocities represented as vectors A and B are combined, the tip of vector B is placed at the tail of vector A, giving a vector whose tail starts where A began and whose tip ends where B ends. If vectors A and B are in a straight line but opposite in direction, they can cancel out, resulting in a zero magnitude, which leads us to the next topic of discussion: vector cancellation.

The velocity of an object is a vector quantity, which means it has both magnitude and direction. Magnitude refers to the speed of the object — how fast it is moving — while direction specifies which way the object is moving.

Expressing velocity requires both components; for instance, if a car is traveling at 50 kilometers per hour towards the north, both the speed (50 km/h) and the direction (north) fully describe its velocity. Changing either the magnitude (speed) or direction of the object's movement will result in a different velocity vector.

When we calculate the resultant velocity of two separate velocities, we are effectively finding a new vector that incorporates both magnitudes and directions of the original velocities. This idea is pivotal when trying to understand motion in two dimensions, such as a plane flying with wind influence or a boat moving across a river current. The magnitude and direction of the resultant velocity provide a clear comprehension of the overall motion.

Vector cancellation happens when two vectors of equal magnitude but opposite direction are added together. Their combined effect results in a resultant vector of zero magnitude — essentially, they nullify each other's influence.

This phenomenon is quite common in physics problems that deal with opposing forces or movements. As illustrated in the given exercise, when two velocities of equal speed but opposite direction are considered — one to the right and one to the left — the addition of these vectors leads to cancellation. They produce a resultant velocity of zero.

This situation could be analogous to two equally strong people pushing on opposite ends of a sofa with equal force; the sofa would not move because the opposing forces cancel each other out. It's important to note that this kind of vector cancellation is specific to vectors that are directly opposed and share the same line of action.