Two points are given: A (4; 0; 5) and B (3; 2; -1) Calculate the coordinates of AB and its length.

1. Let’s define the coordinates of the vector AB. The coordinates of the vector with the beginning A (x1; y1; z1) and the end B (x2; y2; z2) are numbers: a1 = x2 – x1, a2 = y2 – y1 and a3 = z2 – z1 (subtract the beginning from the end).
Let’s find a1:
a1 = x2 – x1 = 3 – 4 = -1.
Let’s find a2:
a2 = y2 – y1 = 2 – 0 = 2.
Let’s find a3:
a3 = z2 – z1 = -1 – 5 = – 6.
Thus, the coordinates of the vector AB (-1; 2; -6).
2. The length of the vector is calculated by the formula:
| AB | = √ (a1 ^ 2 + a2 ^ 2 + a3 ^ 2) = √ ((- 1) ^ 2 + 2 ^ 2 + (-6) ^ 2) = √ (1 + 4 + 36) = √41.
Answer: AB (-1; 2; -6); | AB | = √41.