Can the sum of two unit vectors be a unit vector? Can the difference?

To determine whether the sum or difference of two unit vectors can be a unit vector, let's start by defining what a unit vector is.

A unit vector is a vector that has a magnitude of 1. In other words, it represents a direction or orientation without any change in length. A unit vector can be formed by dividing any vector by its magnitude.

Now, let's consider the sum of two unit vectors. If we add two unit vectors, the resulting vector will have a magnitude larger than 1 unless the two unit vectors have the same direction. In other words, if two unit vectors are parallel and pointing in the same direction, their sum will also be a unit vector. For example, if we add a unit vector pointing to the right (1, 0) and another unit vector pointing to the right (1, 0), their sum will be a unit vector (2, 0) in the same direction. However, if the two unit vectors have different directions, their sum will have a magnitude larger than 1. Therefore, the sum of two unit vectors can only be a unit vector if they are parallel and pointing in the same direction.

On the other hand, when we subtract two unit vectors, the resulting vector can be a unit vector regardless of their directions. This is because subtracting a vector from another effectively changes its direction but not its magnitude. Therefore, the difference of two unit vectors can be a unit vector. For example, if we subtract a unit vector pointing to the right (1, 0) from a unit vector pointing to the left (-1, 0), their difference vector will be a unit vector pointing down (0, -1).

In summary, the sum of two unit vectors can only be a unit vector if they are parallel and pointing in the same direction. The difference of two unit vectors can be a unit vector regardless of their directions.