#### You may also like A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular? As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX. ### Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

# Bond Angles

##### Age 16 to 18Challenge Level

This problem makes use of vectors, coordinates and scalar products. It is a very useful fact that if two vectors ${\bf u}$ and ${\bf v}$ have coordinates $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ then the cosine of the angle $\theta$ they make is $$\cos(\theta) = \frac{{\bf u}\cdot {\bf v}}{|{\bf u}||{\bf v}|}\quad\,,\mbox{ where } {\bf u}\cdot {\bf v}=x_1x_2+y_1y_2+z_1z_2$$
This is useful, because if we can label the coordinates of the atoms in a molecule or crystal then we can easily work out the angles between the bonds.

 In a perfect tetrahedral molecule there is a central atom attached to four other atoms which lie on the vertices of a perfect tetrahedron. If the central atom is at the coordinate origin and the molecule rests on the plane $z=-h$ then what would be the coordinates of the other atoms, assuming a bond length of $1$ unit? Find the scalar products between the vectors joining the origin to each atom and hence the value of $h$. It is usually stated that the angles each of these bonds make is $109.5^\circ$; however, this is only an approximation. What is the exact value of the bond angle in a perfect tetrahedron? If the perfect tetrahedron is deformed slightly, how many of the bond angles could be exactly $109.5^\circ$? How would it be deformed to achieve this? Would the other bond angles increase or decrease under such a deformation? A perfect trigonal pyramidal molecule is the same as a perfect tetrahedral molecule with a single outer atom removed. Ammonia NH$_3$ is approximately a trigonal pyramid with bond angle $107.5$. Does this correspond to a lengthening or shortening of the bonds relative to a perfect structure? By what percentage? If the H atoms were fixed and a vertical force were applied downwards to the N atom in the diagram, would the bond angles increase or decrease? What would be the maximum possible angle mathematically? 