Question # 94346

대답:

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

설명:

두 벡터가된다. #vec (AB) # 과 #vec (AC) #:

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

AC = (x_ (AB) x_ (AC)) + (y_ (AB) y_ (AC)

우리는:

# P = (1; 1; 1) #

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

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

따라서

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

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

과

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

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

따라서:

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

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

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

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