Given: A (-3; 1; -1) B (2; -4; 1) find the vector AB and the modulus of the vector AB.

We are given the coordinates of points A (-3; 1; -1) and B (2; -4; 1). And we need to find the vector AB and the modulus of the vector AB.

To find the vector AB, we must subtract the coordinates of the beginning of the segment from the coordinates of the end of the vector.

So, we find the coordinates of the vector AB:

AB = (2 – (-3); -4 – 1; 1 – (-1)) = (5; -5; 2).

It remains for us to find the module of the vector. It is equal to the square root of the sum of the squares of its coordinates.

So, we calculate:

| AB | = √ (5 ^ 2 + (-5) ^ 2 + 2 ^ 2) = √ (25 + 25 + 4) = √54.

Answer: AB (5; -5; 2), | AB | = √54.