Addition of Vectors

The vector is the quantity that combines the duo- magnitude and direction. Vectors are depicted by the direct pointed line in which the length shows the vector and magnitude, and the orientation shows the direction of the vector. The addition of vectors cannot be done directly; their summing is not as easy as the addition of scalars. Let us consider an example of a bus traveling 20 miles East and 20 miles West to clear this point. So, the total distance traveled is around 40 miles, but displacement is obviously zero. The East and West displacements are both vector quantities, and the opposite directions can cause the individual different displacements to easily cancel each other out. We will learn better about the addition of vectors further in this article.

Addition of vectors by triangle method

The triangle law is often used for the addition of vectors. This method is also named the head-to-tail method. According to this law, two vectors can be added together by placing them together in a way that the former vector’s head joins the tail of the latter vector. We can obtain the resultant sum vector if we match the first’s tail to the second’s head. The addition of vectors using the triangle method can be with the following steps:

Suppose there are two vectors: p and q

Now, draw a line PQ representing p with P as the tail and Q  as the head. Draw another line ST representing q with S as the tail and T as the head. Now join the line PT with P as the tail and T as the head. The line PT represents the resultant sum of the vectors p and q.

The line PT represents p + q.

The magnitude of p + q is: 

√p2 + q2 + 2pq cos θ

Where,

p= magnitude of the vector

q= magnitude of the vector

θ= angle between p and q.

Let the resultant make an angle of ϕ with p then:

 =q  p+q cos

We can also learn this addition of vectors with an example. Consider that we have two vectors with equal magnitude P, and θ is the angle between these two vectors. To look for the magnitude and direction of the resultant, we will use the formulas mentioned above.

Suppose Q is the magnitude of the resultant, then the expression for this is:

Q = √P2 + P2 + 2PP cos θ  = 2 P cos θ/2

Let’s say that the resultant vector makes an angle Ɵ with the first vector p; then the expression will be:

Tan ϕ = P sinθ / P + Pcosθ = tan θ/2

Or,

Ɵ = θ/2

Addition of vectors by parallelogram method

We can learn the addition of vectors through the Parallelogram method, and we can also try adding vectors graphically. According to it, if two vectors are co-existent at a point and are depicted in magnitude and direction by the two sides of a parallelogram drawn from a point; Their addition is given by the diagonal of that parallelogram with magnitude and direction passing through the same point. 

According to it, if two vectors, p and q, denote two adjacent sides of a parallelogram, both pointing outwards, the diagonal drawn through the junction of the two vectors represent the addition. Its magnitude is shown by the square of the diagonal of the parallelogram, equal to the sum of the square of the adjacent sides.

Addition of vectors by polygon method or addition of two vectors

The Polygon method is also a unique way to add two vectors. When sides of a polygon are taken in the same order, their addition is represented by magnitude and direction if the number of vectors can be shown in magnitude and direction. The ending side of the polygon is taken in the opposite direction.

Let A, B,  C, and D → be the four vectors whose sum we will find out.

Consider triangle OKL, in which the vectors A, and B, are denoted as sides OK, KL and are taken simultaneously. We have already learnt about the triangle method of addition of vectors; from that, we concluded that the closing side KL  has to be in the opposite direction so that it represents the summed vector  OR and  KL

Hence, we come to- 

OK+  KL= OL  (Taking it as equation 1.) 

From the triangle method of addition of vectors, triangle OLM can be written as OM

is the resultant of the vectors  OL and  LM. Which is,

OL + LM =→ LM  (What we derived from equation 1.) 

OK+  KL+ LM  = OM (Taking it as equation 2.)

If we apply the triangle method of addition of vectors to triangle OMN,

OM + MN ON 

What we derived from equation 2.

OK+  KL LM + MN = ON (Taking it as equation 3.)

Hence, OK= A KL , B LM, C MN, D 

Considering  ON= R, 

The equation is now

A+ B+ C+ D = R .

Conclusion

It should be kept in mind that vectors are not added algebraically. Geometrical methods sum them.Note that vectors whose resultant have to be calculated always behave independently.The addition of vectors is finding the resultant of several vectors acting on a body.In addition to vectors, the resultant vector is independent of the order of vectors.