Derivation of Cosine and Sine Method of Vector Sum

[bagasme]

Hello all,

In high school physics, the magnitude sum of vector addition can be found by cosine rule:

$$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$

and its angle are calculated by sine rule:

$$\frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1} {\sin ~ (\alpha - \beta)} = \frac {\vec F_2} {\sin ~ \beta}$$

where $\alpha$ is the angle between two vectors, and $\beta$ is the angle between $\vec F_1$ and $\vec R$.

In undergraduate physics books, however, the methods above are not taught, instead the vector addition is done by components, and use $tan ~ \theta = \frac {R_y} {R_x}$ to obtain the angle.

How are derivations of vector addition formulas by cosine and sine rule (as above), and in what cases those formulas can be used in place of vector addition by components?

Bagas

 

[Ibix]

Draw the vectors as arrows. Place the tail of one arrow at the tip of the other. This forms two sides of a triangle. The vector sum is the third side of the triangle, with its head at the head of the second vector.

You now have a triangle with two known sides and one known angle. Any method you like to find the length and angle of the third side is acceptable. Looks like your high school textbook preferred classical geometry. A lot of physicists almost always just resolve the vectors into components and prefer that approach. Neither approach is wrong - its just personal preference and judgement based on exactly what knowns and unknowns you have.

 

[PeroK]

In undergraduate physics books, however, the methods above are not taught ...

You'll always need the rule of cosines and the rule of sines. They are generally useful at all levels of physics.

 

[vanhees71]

Note that first of all one has to guess the meaning of your symbols and 2nd most probably the formulae you quote are wrong.

I guess you want to show the cosine and sine theorems for triangles. Consider an arbitrary triangle with points $A$, $B$, and $C$.

The cosine rule is most simple to derive. For that you only need
$$\overrightarrow{AB} + \overrightarrow{BC}=\overrightarrow{AC} \; \Rightarrow \; \overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB}.$$
Taking the square in the sense of the scalar product of this yields
$$|BC|^2=|AC|^2 + |AB|^2 -2 \overrightarrow{AC} \cdot \overrightarrow {AB}=|AC|^2 + |AB|^2 - 2 |AC| |AB| \cos \alpha,$$
where $\alpha$ is the angle at the point $A$. This is the cosine rule.

The sine rule is most easily derived by calculating the area of the triangle with help of the cross product. You can use any two of the vectors making up the triangle you like:
$$2F=|\overrightarrow{AB} \times \overrightarrow{AC}|=|AB| |AC| \sin \alpha$$
or
$$2F=|\overrightarrow{BA} \times \overrightarrow{BC}|=|AB| |BC| \sin \beta$$
Since both are equal you get
$$|AC| \sin \alpha =|BC| \sin \beta \; \Rightarrow \; \frac{|AC|}{\sin \beta} = \frac{BC}{\sin \alpha},$$
which is the sine rule.

 

[Vanadium 50]

In undergraduate physics books, however, the methods above are not taught

Neither is counting on your fingers and toes.