Question

Question #94346

Answer 1

`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)`