Stewart 8e 12.2.55
Find the angle between a diagonal of a cube and one of its edges.solutionSince the size is irrelevant to this angle, consider a unit cube in the first octant of 3-space where all three coordinates are zero or positive. We can draw a unit cube using a plotcommand that bob remembered about (not necessary, but it always helps to have a diagram when possible):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 main diagonal has position 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Its length is just LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbXNxcnRHRiQ2Iy1GIzYpLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIitGJ0Y6LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlJGNkY9RjYvJStleGVjdXRhYmxlR0ZDRjpGVUY6RitGVUY6so its unit vector is the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElZGhhdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkmbWZyYWNHRiQ2KC1GLDYlUSJkRidGL0YyLUYjNictRiw2I1EhRictRiM2JkZXLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW5HRiQ2JFEiM0YnRjkvJStleGVjdXRhYmxlR0Y9RjlGX29GOUZXRl9vRjkvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZvLyUpYmV2ZWxsZWRHRj1GX29GOQ==The first edge is along the x axis and the unit vector of its direction 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Then dotting the two unit vectors gives the cosine of the angle between them, which we can solve for the radian measure of the angle and convert to degrees: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So the angle is about 55 degrees. The radian measure is not very useful since we have no feeling for its values:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEnYXJjY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiNRIUYnLUYjNigtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnRjItRkM2JFEiM0YnRjIvJS5saW5ldGhpY2tuZXNzR0ZFLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdGMS1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGYG8tRiM2JS1JJm1zcXJ0R0YkNiNGRi8lK2V4ZWN1dGFibGVHRjFGMkY6RmhvRjJGOkZob0YyRjItRlM2LVEiOkYnRjJGVkZYRlpGZm5GaG5Gam5GXG8vRl9vUSwwLjI3Nzc3NzhlbUYnL0Zib0ZecC1GUzYtUSJ+RidGMkZWRlhGWkZmbkZobkZqbkZcb0Zeb0Zhby1GLDYlUSZldmFsZkYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLUY2NiQtRiM2Jy1GLDYlUSIlRidGZnBGaHAtRlM2LVEiLEYnRjJGVi9GWUZncEZaRmZuRmhuRmpuRlxvRl5vL0Zib1EsMC4zMzMzMzMzZW1GJy1GQzYkUSI1RidGMkZob0YyRjJGaG9GMg==We can display the arrow vectors in the box: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Stewart 8e 12.2.57Tetrahedron molecule LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEjQ0hGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JVEiNEYnRjUvRjlRJ25vcm1hbEYnRjVGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGQQ==, vertices (H) at points 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; centroid (C) at centroid: 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. Find the angle between two vertices through the centroid: the H-C-H bond angle.solutionThe position of the C atom at the centroid is shown by the red arrow.[Removing the number sign from the next line and executing it gives the help page on polygon.]LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEpP3BvbHlnb25GJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRicvJStleGVjdXRhYmxlR0Y0Ri8=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 vectors from C to two H locations: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 make arrows along these bonds from the Carbon location.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 along the black vectors (bond directions)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 angle between them 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 bond angle is about 109.5 degrees, about 20 degrees past a right angle. It looks about right.