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Karsten's answer is excellent, but here is a figure that shows the mathemathics involved:

The central atom (green) is at the center of the cube, the four other atoms (purple) are at alternating vertices and the geometry should be clear.

Alternatively, if you orient the molecule so one peripheral (purple) atom is directly "above" the central (green) atom, then each of the other three atoms is just $1 - \theta$ ($\approx 70.52877940 ^\circ$) away from being directly "under" the central atom, so each contributes $\cos(1 - \theta)$ times the bond dipole. This is the downward component of the bond dipole from one of the lower atoms.

But $\cos(1 - \theta) = 1/3$, so this is simply (bond dipole)/3, and there are three of these lower atoms, so the three downward components exactly balance the one upward component.

Karsten's answer is excellent, but here is a figure that shows the mathematics involved:

The central atom (green) is at the center of the cube, the four other atoms (purple) are at alternating vertices and the geometry should be clear.

Alternatively, if you orient the molecule so one peripheral (purple) atom is directly "above" the central (green) atom, then each of the other three atoms is just $1 - \theta$ ($\approx 70.52877940 ^\circ$) away from being directly "under" the central atom, so each contributes $\cos(1 - \theta)$ times the bond dipole. This is the downward component of the bond dipole from one of the lower atoms.

But $\cos(1 - \theta) = 1/3$, so this is simply (bond dipole)/3, and there are three of these lower atoms, so the three downward components exactly balance the one upward component.

Karsten's answer is excellent, but here is a figure that shows the math involved:

The central atom (green) is at the center of the cube, the four other atoms (purple) are at alternating vertices and the geometry should be clear.

Alternatively, if you orient the molecule so one peripheral (purple) atom is directly "above" the central (green) atom, then each of the other three atoms is just 1 - $\theta$ (about 70.52877940 degrees) away from being directly "under" the central atom, so each contributes cos(1 - $\theta$) times the bond dipole. This is the downward component of the bond dipole from one of the lower atoms.

But cos(1 - $\theta$) = 1/3, so this is simply (bond dipole)/3, and there are three of these lower atoms, so the three downward components exactly balance the one upward component. Hope this helps!

Karsten's answer is excellent, but here is a figure that shows the mathemathics involved:

The central atom (green) is at the center of the cube, the four other atoms (purple) are at alternating vertices and the geometry should be clear.

Alternatively, if you orient the molecule so one peripheral (purple) atom is directly "above" the central (green) atom, then each of the other three atoms is just $1 - \theta$ ($\approx 70.52877940 ^\circ$) away from being directly "under" the central atom, so each contributes $\cos(1 - \theta)$ times the bond dipole. This is the downward component of the bond dipole from one of the lower atoms.

But $\cos(1 - \theta) = 1/3$, so this is simply (bond dipole)/3, and there are three of these lower atoms, so the three downward components exactly balance the one upward component.

Karsten's answer is excellent, but here is a figure that shows the math involved:

The central atom (green) is at the center of the cube, the four other atoms (purple) are at alternating vertices and the geometry should be clear.

Alternatively, if you orient the molocule so one peripheral (purple) atom it directly "above" the central (green) atom, then each of the other three atoms is just 1 - $\theta$ (about 70.52877940 degrees) away from being directly "under" the central atom, so each contributes cos(1 - $\theta$) times the bond dipole. This is the downward component of the bond dipole from one of the lower atoms.

But cos(1 - $\theta$) = 1/3, so this is simply (bond dipole)/3, and there are three of these lower atoms, so the three downward components exactly balance the one upward component. Hope this helps!

Karsten's answer is excellent, but here is a figure that shows the math involved:

The central atom (green) is at the center of the cube, the four other atoms (purple) are at alternating vertices and the geometry should be clear.

Alternatively, if you orient the molecule so one peripheral (purple) atom is directly "above" the central (green) atom, then each of the other three atoms is just 1 - $\theta$ (about 70.52877940 degrees) away from being directly "under" the central atom, so each contributes cos(1 - $\theta$) times the bond dipole. This is the downward component of the bond dipole from one of the lower atoms.

But cos(1 - $\theta$) = 1/3, so this is simply (bond dipole)/3, and there are three of these lower atoms, so the three downward components exactly balance the one upward component. Hope this helps!