The coordinates of points A, B, C are given: A (7; -4; 1), B (12; -3; 1), C (10; 1; 5). Find: 1) coordinates of vectors AB

The coordinates of points A, B, C are given: A (7; -4; 1), B (12; -3; 1), C (10; 1; 5). Find: 1) coordinates of vectors AB and AC; 2) the lengths of the vectors AB and AC; 3) the angle between vectors AB and AC.

1) To find the coordinates of a vector, you need to find the difference between the corresponding coordinates of the end point of the vector and the beginning.

Let’s find the coordinates of the vector AB:

AB (xv – xa; yv – ya; zv – za);

AB (12 – 7; -3 – 4; 1 – 1);

AB (5; -7; 0).

Let’s find the coordinates of the AC vector:

AC (xc – ha; us – ya; zc – za);

AC (10 – 7; 1 – 4; 5 – 1);

AC (3; -3; 4).

2) The square of the length of a vector is equal to the sum of the squares of its coordinates.

Let’s find the length of the vector AB:

| AB | 2 = 52 + (-7) 2 + 02 = 25 + 49 = 74;

| AB | = √74.

Let’s find the length of the AC vector:

| AC | 2 = 32 + (-3) 2 + 42 = 9 + 9 + 16 = 34;

| AU | = √34.

3) Find the angle between the vectors AB and AC.

cosBAC = (5 * 3 + (-7) * (-3) + 0 * 4) / (√74 * √34) = 36 / √2516 = 9 / √629

angle BAC = arcos (9 / √629)

angle BAC = 690