Domanda n. 94346

Risposta:

#hat (PQR) = cos ^ (- 1) (27 / sqrt1235) #

Spiegazione:

Sii due vettori #vec (AB) # e #vec (AC) #:

#vec (AB) * vec (AC) = (AB) (AC) cos (hat (BAC)) #

# = (X_ (AB) x_ (CA)) + (y_ y_ (AB) (AC)) + (z_ (AB) z_ (CA)) #

Abbiamo:

# P = (1; 1; 1) #

#Q = (- 2, 2, 4) #

# R = (3; -4; 2) #

perciò

#vec (QP) = (x_P-x_Q; y_P-y_Q; z_P-z_Q) = (3; -1; -3) #

#vec (QR) = (x_R-x_Q; y_R-y_Q; z_R-z_Q) = (5; -6; -2) #

e

# (QP) = sqrt ((x_ (QP)) ^ 2 + (y_ (QP)) ^ 2 + (z_ (QP)) ^ 2) = sqrt (9 + 1 + 9) = sqrt (19) #

# (QR) = sqrt ((x_ (QR)) ^ 2+ (y_ (QR)) ^ 2+ (z_ (QR)) ^ 2) = sqrt (25 + 36 + 4) = sqrt (65) #

Perciò:

#vec (QP) * vec (QR) = sqrt19sqrt65cos (cappello (PQR)) #

#=(3*5+(-1)(-6)+(-3)(-2))#

#rarr cos (hat (PQR)) = (15 + 6 + 6) / (sqrt19sqrt65) = 27 / sqrt1235 #

#rarr hat (PQR) = cos ^ (- 1) (27 / sqrt1235) #