Dado el segmento de extremos A(2, 1, -1) y B(5, -2, 8): a. Calcula las coordenadas del punto C, de forma que B sea el punto medio del segmento AC b. Calcula las coordenadas de dos puntos P y Q pertenecientes al segmento AB tales que dividan a este en tres segmentos de igual longitud.
wrong again.
a) B-A = (5, -2, 8) - (2, 1, -1) = (3,-3,9)
Since V is the midpoint of AC, C-B = B-A
so C = B + (A-B) = (5, -2, 8)+(3,-3,9) = (8,-5,17)
b) So now we know that C-A = (6,-6,18)
now we want 1/3 of that, which is x=(2,-2,6)
P = A+x = (4,-1,5)
Q = P+x = (6,-3,11)
a. Para calcular las coordenadas del punto C, utilizamos la fórmula del punto medio:
C = ((A + B) / 2)
Donde A y B son los extremos del segmento.
C = ((2+5)/2, (1-2)/2, (-1+8)/2)
C = (7/2, -1/2, 7/2)
Por lo tanto, las coordenadas del punto C son (7/2, -1/2, 7/2).
b. Para calcular las coordenadas de los puntos P y Q que dividen al segmento AB en tres segmentos de igual longitud, utilizamos la fórmula:
P = (2/3)A + (1/3)B
Q = (1/3)A + (2/3)B
P = (2/3)(2, 1, -1) + (1/3)(5, -2, 8)
= (4/3, 2/3, -2/3) + (5/3, -2/3, 8/3)
= (9/3, 0, 6/3)
= (3, 0, 2)
Por lo tanto, las coordenadas del punto P son (3, 0, 2).
Q = (1/3)(2, 1, -1) + (2/3)(5, -2, 8)
= (2/3, 1/3, -1/3) + (10/3, -4/3, 16/3)
= (12/3, -3/3, 15/3)
= (4, -1, 5)
Por lo tanto, las coordenadas del punto Q son (4, -1, 5).
I apologize for the mistake. You are correct. Here is the correct solution:
a) To find the coordinates of point C such that B is the midpoint of segment AC, we can use the midpoint formula:
C = (A + B) / 2
C = ((2, 1, -1) + (5, -2, 8)) / 2
= (7, -1, 7) / 2
= (7/2, -1/2, 7/2)
Therefore, the coordinates of point C are (7/2, -1/2, 7/2).
b) To find two points P and Q that divide segment AB into three equal segments, we can use the following formulas:
P = A + (1/3)(B - A)
Q = A + (2/3)(B - A)
P = (2, 1, -1) + (1/3)((5, -2, 8) - (2, 1, -1))
= (2, 1, -1) + (1/3)(3, -3, 9)
= (2, 1, -1) + (1, -1, 3)
= (3, 0, 2)
Therefore, the coordinates of point P are (3, 0, 2).
Q = (2, 1, -1) + (2/3)((5, -2, 8) - (2, 1, -1))
= (2, 1, -1) + (2/3)(3, -3, 9)
= (2, 1, -1) + (2, -2, 6)
= (4, -1, 5)
Therefore, the coordinates of point Q are (4, -1, 5).