Question #94346

1 Answer
Jul 1, 2015

Answer:

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

Explanation:

Be two vectors #vec(AB)# and #vec(AC)#:

#vec(AB) * vec(AC) = (AB)(AC)cos(hat(BAC))#
#=(x_(AB)x_(AC))+(y_(AB)y_(AC))+(z_(AB)z_(AC))#

We have:

#P=(1;1;1)#
#Q=(-2;2;4)#
#R=(3;-4;2)#

therefore

#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)#

and

#(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)#

Therefore:

#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)#