#### IIT (NTSE/Olympiad)

# Vector

__Vector Addition__

__Vector Addition__

**3. VECTOR ADDITION**

There are two methods for addition of vectors :

3.1 Graphical method

3.2 Mathematical method

**3.1 Graphical Method :**

**(i) Triangle rule (Used to add two vectors only) :**

If and are the two vectors to be added, a diagram is drawn in which the tail of coincides with the head of . The vector joining the tail of with the head of is the

**vector sum**of and . Figure shows the construction.

**(ii) Polygon method : (used to add more than two vectors)**

We use this method for more than two vector. Suppose , , are three vectors to be added. A diagram is drawn in which the tail of coincides with the head of and tail of coincides with head of . The vector joining the tail of and head ofis called the

**resultant vector**and this is the vector sum of three given vectors ( + + ).

**Result : 1.**

If three or more vectors themselves complete a triangle or a polygon, then their sum-vector cannot be drawn. It means that the sum of these vectors is zero.

**Result : 2.**

If there are two vectors a1 and a2 with equal magnitude, then the resultant of their addition will bisect the angle between them.

**Result : 3.**

If we add two different vectors

**a**with equal magnitude and angle between them is 120º, then the resultant would bisect the angle and magnitude would be equal to each of the magnitude of vector.

_{1}and a_{2}**3.2 Mathematical method :**

**3.2.1 For two vectors :**

If two vectors and makes angle θ to each other than the magnitude of their vector addition

and if the resultant vector makes the angle α with the vector then it is given by

tan α =

and if the resultant vector makes the angle with the vectorthen it is given by

tan β =

and R

^{2}= (a + b cos θ )

^{2}+ (b sin θ )

^{2}

Where θ is the angle between a and b.

**3.2.2 For more then two vectors (Component method) :**

and R = + + = (a

_{x}+ b

_{x}+ c

_{x}) + (a

_{y}+ b

_{y}+ c

_{y}) + (a

_{z}+ b

_{z}+ c

_{z})

Suppose a

_{z}= b

_{z}= c

_{z}= 0 and

= a

_{x}+ a

_{y}

= b

_{x}– b

_{y}

= – c

_{x}+ c

_{y}

So the resultant = R

_{x}+ R

_{y}= (a

_{x}+ b

_{x}– c

_{x}) + (a

_{y}– b

_{y}+ c

_{y})

#### IIT (Class X)

- Unit, Dimension & Error

- Vectors

- Motion in One Dimension

- PROJECTILE MOTION

- NEWTON'S LAWS OF MOTION & FRICTION

- WORK, POWER, ENERGY & CONSERVATION LAWS

- CIRCULAR MOTION & ROTATIONAL DYNAMICS

- GRAVITATION