Laws of Vector Addition
There are three basic laws of vector addition that are used to add vectors and that include,
- Triangle Law of Vector Addition
- Parallelogram Law of Vector Addition
- Polygon Law of Vector Addition
Let’s understand these laws of vector addition in detail as follows:
Triangle Law of Vector Addition
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 3rd side of that triangle taken in opposite order.
Read more about Triangle Law of Vector Addition.
Derivation of Triangle 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 the 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,
OB2=OC2 + BC2
⇒OB2 = (OA + AC)2 + BC2 . . . (i)
In triangle ACB,
cos θ = AC/AB
⇒ AC = AB cos θ = Q cos θ
Also, sin θ = BC/AB
⇒ BC = AB sin θ = Q sin θ
Magnitude of Resultant Vector
Substituting the value of AC and BC in (i), we get
R2 =(P + Q cos θ)2 + (Q sin θ)2
⇒R2 = P2 + 2 PQ cos θ + Q2 cos2 θ + Q2sin2 θ
⇒ R2 = P2 + 2PQ cos θ + Q2
[Tex] \therefore R=\sqrt{P^2 + 2PQ \cos \theta + Q^2} [/Tex]
Which is the magnitude of the resultant.
Direction of Resultant Vector
Let Φ be the angle made by resultant R with P. Then,
From triangle OBC,
tan Φ = BC/OC = BC/(OA + AC)
⇒ tan Φ = Q sin θ/(P + Q cos θ)
[Tex]\therefore \phi = tan ^ {-1} ( \frac {Q sin\theta} {P+Q cos\theta} ) [/Tex]
Which is the direction of the resultant.
Vector Addition
Vector Addition in Mathematics is the fundamental operation of vector algebra that is used to find the sum of two vectors. Vectors are mathematical quantities that have magnitude and direction. A vector can be represented by a line with an arrow pointing towards its direction and its length represents the magnitude of the vector.
Vector addition is achieved by taking the vector in 3D or 2D and then arranging them such that, the head of one vector is arranged touching the tail of the other vector and now a third vector joins the tail of the first vector with the head of the other vector gives the sum of the vectors.
In this article, we will learn about, vector definition, vector addition, laws of vector addition, and others in detail.
Table of Content
- What is Vector?
- What is Vector Addition?
- Vector Addition Calculator
- Laws of Vector Addition
- Triangle Law of Vector Addition
- Parallelogram Law of Vector Addition
- Polygon Law of Vector Addition
- Vector Addition Formula
- Properties of Vector Addition
- Vector Subtraction
- Summary: Vector Addition
- Examples of Addition of Vectors