שאלה # 94346

תשובה:

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

הסבר:

להיות שני וקטורים #vec (AB) # ו #vec (AC) #:

# #cc (AB) * vec (AC) = (AB) (AC) cos (האט (BAC)) #

# (x (AB) x_ (AC)) + (y_ (AB) y_ (AC)) + (z_ (AB) z_ (AC)) #

יש לנו:

# P = (1, 1; 1) #

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

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

לכן

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

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

ו

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

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

לכן:

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

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

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

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