Parallelogram Vector addition

One of the most common vector operations that is frequently encountered is addition operation. Addition of two or more vectors to arrive to a vector sum is called vector addition. Consider two vectors,  u=(u1, u2) and  v=( v1, v2), the sum of these vectors would be vector u + vector v= (u1+v1, u2+v2). This sum is called the resultant vector. There are various methods to find the resultant vector namely, parallelogram method, component method, graphical method, cosine method, polygon method etc.  

Parallelogram Vector addition:  In this method first two vectors are drawn such that their initial points coincide. Then the other two lines are drawn to form a parallelogram. The resultant would be the diagonal of the parallelogram drawn from the initial point to the opposite vertex of the parallelogram.

Vector addition component method is one way used in adding vectors. Component means ‘part ’and hence they can be considered as the coordinates of the point that is associated with the vector. In a Euclidean plane consider two vectors, u=(u1, u2) and v=( v1, v2), the resultant vector which is the sum of these vectors is given by, u+v = (u1+v1, u2+v2). In a three dimensional space, given vectors u=(u1, u2, u3) and v=( v1, v2,v3) the method would be similar to the method used in addition of vectors in a Euclidean plane. So, u+v = (u1+v1, u2+v2, u3+v3). We can finally conclude that vector-addition is just like the normal addition, component by component.

Let us now learn the vector addition graphical method, consider two vectors, u=(4,3) and v=(1,4) in the plane. Using the component method of vector-addition the sum can be given as, u+v = (4+1, 3+4) = (5,7). Using the graphical method we get the same resultant vector by taking one vector whose direction and magnitude is unchanged and placing its end at the unchanged vector’s tip, and joining the origin and the new location of the displaced vector using an arrow. This procedure in general works for addition of vectors. For any two given vectors u and v in the plane, the sum of the vectors in general can be graphically represented as the vector addition diagram given below

As the vectors in a two dimensional space lie in the same plane, any two vectors in a three dimensional space also lie in the same plane and hence graphical method works well for vector-addition in a 3-dimensional space.