IIT (NTSE/Olympiad)  

Vector

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 a1 and a2 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.
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 R2 = (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 = + + = (ax + bx + cx) + (ay + by + cy) + (az + bz + cz)
Suppose az = bz = cz = 0 and
= ax + ay
= bx – by
= – cx + cy
So the resultant = Rx + Ry = (ax + bx – cx) + (ay – by + cy)


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