# Vectors Algebra – Sine Rule

## Vectors Algebra – Sine Rule

Sine Rule: If A, B and C are the vertices of a triangle ABC, then sine rule $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$,

Let $$\overrightarrow{BC}=\overrightarrow{a}$$,

$$\overrightarrow{CA}=\overrightarrow{b}$$,

$$\overrightarrow{AB}=\overrightarrow{c}$$,

So that $$\overrightarrow{a}+\overrightarrow{b}=-\overrightarrow{c}$$,

Therefore   $$\overrightarrow{a}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{a}\times \overrightarrow{c}$$,

$$\left( \because \overrightarrow{a}\times \overrightarrow{a}=0\ ,-\overrightarrow{a}\times \overrightarrow{c}\ =\overrightarrow{c}\times \overrightarrow{a}\ \right)$$,

$$0+\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{a}$$,

$$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{a}$$,

$$\left| \overrightarrow{a}\times \overrightarrow{b} \right|=\left| \overrightarrow{c}\times \overrightarrow{a} \right|$$,

$$\left( \because|\overrightarrow{a}\times \overrightarrow{b}|=a.b\sin \theta \ \right)$$,

$$a.b\ \sin ({{180}^{o}}-C)=c.a\sin ({{180}^{o}}-B)$$,

$$\left( \because \sin ({{180}^{o}}-\theta )\ =\sin \theta \right)$$,

$$a.b\ \sin (C)=c.a\sin (B)$$,

Dividing both sides by abc, we get

$$\frac{a.b\ \sin C}{abc}=\frac{c.a\sin B}{abc}$$,

$$\frac{\ \sin C}{c}=\frac{\sin B}{b}$$ . . .(1)

$$\frac{b}{\sin B}=\frac{c}{\sin C}$$, . . .(1)

Similarly,   . . .(2) $$\frac{c}{\sin C}=\frac{a}{\sin A}$$, . . .(2)

From (1) and (2), we have

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$,

A, B and C are the vertices of a triangle ABC, then sine rule    $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$.