What's the relationship between displacement and distance as vectors and scalars?

Displacement is a vector quantity that considers direction, while distance is a scalar quantity that only considers magnitude.

In physics, both displacement and distance are used to describe the motion of an object. However, they are not the same and have different properties due to their nature as vector and scalar quantities respectively.

Displacement, being a vector quantity, has both magnitude and direction. It is defined as the shortest distance from the initial to the final position of an object. It takes into account the direction of the motion. For example, if you walk 10 metres east and then 10 metres west, your total displacement is zero because you ended up at the same place you started.

On the other hand, distance is a scalar quantity, which means it only has magnitude and no direction. It is the total path covered by an object irrespective of the direction. In the previous example, even though your displacement was zero, the total distance you covered was 20 metres.

This difference becomes particularly important when dealing with problems involving motion in more than one dimension. For example, if an object moves in a circular path and returns to its starting point, its displacement is zero because the initial and final positions are the same. However, the distance travelled is equal to the circumference of the circle.

In summary, displacement and distance are related but distinct concepts in physics. Displacement is a vector quantity that considers the direction of motion and gives the shortest distance between two points, while distance is a scalar quantity that represents the total path covered by an object, regardless of the direction. Understanding the difference between these two is crucial for solving problems in physics, especially those involving motion in multiple dimensions.