# Triangle Law of Vector Addition

## Statement of Triangle Law

If 2 vectors acting simultaneously on a body are represented both in magnitude and direction by 2 sides of a triangle taken in an order then the resultant(both magnitude and direction) of these vectors is given by 3

^{rd}side of that triangle taken in opposite order.

## Derivation of the law

Consider two vectors **P **and **Q** acting on a body and represented both in magnitude and direction by sides OA and AB respectively of a triangle OAB. Let θ be the angle between **P** and **Q**. Let **R** be the resultant of vectors **P** and **Q**. Then, according to triangle law of vector addition, side OB represents the resultant of **P** and **Q**.

So, we have

**R **=** P **+** Q**

Now, expand A to C and draw BC perpendicular to OC.

From triangle OCB,

In triangle ACB,

Also,

**Magnitude of resultant:**

Substituting value of AC and BC in (i), we get

which is the magnitude of resultant.

**Direction of resultant:** Let ø be the angle made by resultant **R **with **P**. Then,

From triangle OBC,

which is the direction of resultant.