(701, #57) A molecule of methane, , is structured with four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by H-C-H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about .
[Hint: Take the vertices of the tetrahedron to be the points (1,0,0), (0,1,0), (0,0,1), and (1,1,1) as shown in the figure. Then the centroid is .

Solution: Let v be the vector from the centroid to the point (0,1,0); let w be the vector from the centroid to the point (0,0,1). Then

Thus if is the angle between v and w then

Ans:

by: gm