Trigonometry/Vectors and Dot Products

Calculating bond angles of a symmetrical tetrahedral molecule such as methane using a dot product

Consider the vectors U and V (with respective magnitudes |U| and |V|). If those vectors enclose an angle θ then the dot product of those vectors can be written as:

$𝐔\cdot 𝐕=|𝐔||𝐕|\cos (\theta )$

If the vectors can be written as:

$𝐔=(U_x,U_y,U_z)$

$𝐕=(V_x,V_y,V_z)$

then the dot product is given by:

$𝐔\cdot 𝐕=U_x{V}_x+U_y{V}_y+U_z{V}_z$

For example,

$(1,2,3)\cdot (2,2,2)=1(2)+2(2)+3(2)=12.$

and

$(0,5,0)\cdot (4,0,0)=0.$

We can interpret the last case by noting that the product is zero because the angle between the two vectors is 90 degrees.

Since

$|𝐔|=\sqrt{U_x^2+U_y^2+U_z^2}$

and

$|𝐕|=\sqrt{V_x^2+V_y^2+V_z^2}$

this means that

$\cos (\theta )=\frac{U_x{V}_x+U_y{V}_y+U_z{V}_z}{\sqrt{U_x^2+U_y^2+U_z^2}\sqrt{V_x^2+V_y^2+V_z^2}}$