Magnitudes y vectores - 2 Vectores 20

Vectores 20. Sean los vectores A =(0, 0, 3), B= (8, 0, 0), C=(0, –2, 0), calcular, en coordenadas cartesianas, los siguientes productos mixtos. Indicar si la respuesta es un escalar o un vector. Graficar en los casos que corresponda.

a) D = (B x C) • (A x C)

(B x C) = (8, 0, 0) x (0, –2, 0) = (0 * 0 - 0 * (- 2) ; 0 * 0  - 8 * 0 ; 8* (- 2) - 0 * 0 ) = (0 ; 0 ; -16)

(A x C) = (0, 0, 3) x (0, –2, 0) = (0 * 0 – 3 * (-2) ; 3 *  0 - 0 * 0 ; 0 * (-2) - 0 *  0 ) = (6 ;  0 ; 0)

(B x C) • (A x C) =  (0 ; 0 ; -16) • (6 ;  0 ; 0) =  0 * 6  + 0 * 0 + (-16) * 0 = 0

D = 0 (escalar)

b) D = –4(B x B) • A – A

(B x B) = (8, 0, 0) x (8, 0, 0) = ( 0 * 0 - 0 * 0 ; 0 * 8 - 8 * 0 ; 8 * 0 - 0 * 8 ) = (0, 0, 0)

–4(B x B) = -4 (0, 0, 0) = (0, 0, 0)

–4(B x B) • A = (0, 0, 0) • (0, 0, 3) = 0 * 0  + 0 * 0 + 0 * 3 = 0

–4(B x B) • A – A = 0 - (0, 0, 3) = (0, 0, -3)

D = (0, 0, -3) (vector)

Nota: No es correcto restar (o sumar) un escalar (–4(B x B) • A ) y un vector (A)

c) D = (A x C) • B

(A x C) = (0, 0, 3) x (0, –2, 0) = (0 * 0 – 3 * (-2) ; 3 *  0 - 0 * 0 ; 0 * (-2) - 0 *  0 ) = (6 ;  0 ; 0)

(A x C) • B = (6 ;  0 ; 0) • (8, 0, 0) = 6 * 8  + 0 * 0 + 0 * 0 = 48

D = 48 (escalar)

d) D =(A x C) • (C x A)

(A x C) = (0, 0, 3) x (0, –2, 0) = (0 * 0 – 3 * (-2) ; 3 * 0 - 0 * 0 ; 0 * (-2) - 0 * 0) = (6 ;  0 ; 0)

(C x A) = (0, –2, 0) x (0, 0, 3) = ((-2) * 3 - 0 * 0 ; 0 * 0 – 0 * 3 ; 0 * 0– (-2) * 0) = (-6 ;  0 ; 0)

(A x C) • (C x A) = (6 ;  0 ; 0) • (-6 ;  0 ; 0) = 6 * (-6) + 0 * 0 + 0 * 0 = -36

D = -36 (escalar)

e) D =(A x B) • (A – B)

A x B = (0, 0, 3) x (8, 0, 0) = (0 * 0 - 3 * 0 ; 3 * 8 - 0 * 0 ; 0 * 0 - 0 * 8 ) = (0 ; 24 ; 0)

(A – B) = (0, 0, 3) - (8, 0, 0) = (-8, 0, 3)

(A x B) • (A – B) = (0 ; 24 ; 0) • (-8, 0, 3) = 0 * (-8) + 24 * 0 + 0 * 3 = 0

D = 0 (escalar)