csat-suneung 2005 Q7

csat-suneung · South-Korea · csat__math-science 3 marks Vectors 3D & Lines Dihedral Angle Computation
As shown in the figure on the right, in a cube ABCD-EFGH with edge length 3, there are three points $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ on the three edges AD, BC, FG such that $\overline { \mathrm { DP } } = \overline { \mathrm { BQ } } = \overline { \mathrm { GR } } = 1$. The angle between plane PQR and plane CGHD is $\theta$. What is the value of $\cos \theta$? (where $0 < \theta < \frac { \pi } { 2 }$) [3 points]
(1) $\frac { \sqrt { 10 } } { 5 }$
(2) $\frac { \sqrt { 10 } } { 10 }$
(3) $\frac { \sqrt { 11 } } { 11 }$
(4) $\frac { 2 \sqrt { 11 } } { 11 }$
(5) $\frac { 3 \sqrt { 11 } } { 11 }$
As shown in the figure on the right, in a cube ABCD-EFGH with edge length 3, there are three points $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ on the three edges AD, BC, FG such that $\overline { \mathrm { DP } } = \overline { \mathrm { BQ } } = \overline { \mathrm { GR } } = 1$. The angle between plane PQR and plane CGHD is $\theta$. What is the value of $\cos \theta$? (where $0 < \theta < \frac { \pi } { 2 }$) [3 points]\\
(1) $\frac { \sqrt { 10 } } { 5 }$\\
(2) $\frac { \sqrt { 10 } } { 10 }$\\
(3) $\frac { \sqrt { 11 } } { 11 }$\\
(4) $\frac { 2 \sqrt { 11 } } { 11 }$\\
(5) $\frac { 3 \sqrt { 11 } } { 11 }$