Cosine rule

The elements G,y can be calculated from the distance matrix using the cosine rule ... 

The geometric properties of vectors can be combined into the triangle relationship, also called the cosine rule, which states that ... 

In particular, and c,-,- represent the sums of squared elements of the row-vectors X, and X,-. Consequently, and d > are the norms (or lengths) of the /th and / -th rows, respectively. The relation that links the three expressions of eq. (31.31) together is called the triangle equation or cosine rule ... 

The metric matrix G was then defined as the matrix where each element g, is the dot product of the two vectors from the center of mass to the points i and From the cosine rule, each element can be calculated ... 

Again, from application of the cosine rule it may be shown that... 

For most energy transfers the incident energy (and hence is much larger than the final energy (and hence k, so the momentum transfer Q is almost equal to k irrespective of the scattering angle. From the Cosine Rule for the solution of triangles, Eq. (2.23) ... 

When the primary electron beam strikes a specimen set at 0 degrees tilt the yield of secondary electrons for different exit trajectories follows a so called cosine rule [lt)=l)00s( )J as shown in figure 3.1.The maximum yield is for electrons with an exit angle normal to the specimen surface with the intensity decreasing as the cosine of the angle the trajectory path makes with the incident beam. 

By applying the cosine rule this can be written as follows (see Figure 4.15) ... 

The distance Loo between the ends of the two segments is given by the cosine rule... 

The vector product is solved with the cosine rule already used in Equation (A4-2)... 

In addition, there is a geometric condition which is obtained by applying the Cosine rule to the triangle BRG,... 

Since the two surfaces are joined together by the trigonometry of the cosine rules this equation is known as the cosine law. 

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