Given points M (1; 3), N (4; 0), K (0; -1) 1) MN = KP vectors, find the coordinate of point P

Given points M (1; 3), N (4; 0), K (0; -1) 1) MN = KP vectors, find the coordinate of point P 2) KM vector perpendicular to the vector NT, T (3; y) find y 3) Find | MN | vector in module 4) find KM * KN vectors 5) find vectors in module | KM-4KN | 6) find cos (angle MKN) 7) write the equation of the lines KM and MN

1) Find the coordinates of the vector MN: ((4 – 1); (0 – 3)) = (3; -3).

Coordinates of the vector KP: ((x – 0); (y – (-1)) = (x; (y + 1)).

Then the coordinates of the point P:

x = 3;

y + 1 = -3;

y = -4.

2) Coordinates of the vector KM: ((1 – 0); (3 – (-1)) = (1; 4). Since the dot product is the product of perpendicular vectors early 0, we get the equation:

3 + 4y = 0;

y = -3/4.

3) MN = ((4 – 1); (0 – 3)) = (3; (-3)).

| MN | = √3 ^ 2 + (-3) ^ 2 = 3√2.

4) KM = (1; 4), KN = (4; 1).

KM * KN = 1 * 4 + 4 * 1 = 8;

5) 4 * KN = (4; 16).