Explain in detail the triangle law of addition.
Answers
1. Triangle law
2. Parallelogram law
But we have to discuss about triangle law , Let's try to understand.
If two vectors can be represented in magnitude and direction by two sides of triangle , then third side of the triangle taken in reverse order represents their sum in magnitude and direction.
Let a vector P is along OA , and a vector Q is along AB , then according to triangle law of addition , R which is along OB is resultant of vectors P and Q
e.g., OB = OA + AB ; R = P + Q .
a rough daigram is shown in figure. You can understand how triangle law of addition works and how we can apply it .
Answer:
Triangle law of vector addition is one of the vector addition laws. Vector addition is defined as the geometrical sum of two or more vectors as they do not follow regular laws of algebra. The resultant vector is known as the composition of a vector.
There are a few conditions that are applicable for any vector addition, they are:
Scalars and vectors can never be added.
For any two scalars to be added, they must be of the same nature. Example, mass should be added with mass and not with time.
For any two vectors to be added, they must be of the same nature. Example, velocity should be added with velocity and not with force.
There are two laws of vector addition, they are:
Triangle law of vector addition
Parallelogram law of vector addition
Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.
Triangle Law of Vector Addition
To obtain R⃗ which is the resultant of the sum of vectors A⃗ and B⃗ with the same order of magnitude and direction as shown in the figure, we use the following rule:
R⃗ =A⃗ +B⃗
You may also want to check out these topics given below!
Vector Addition- Analytical Method
Addition And Subtraction Of Vectors
Triangle Law of Vector Addition Derivation
Consider two vectors P⃗ and Q⃗ that are represented in the order of magnitude and direction by the sides OA and AB, respectively of the triangle OAB. Let R⃗ be the resultant of vectors P⃗ and Q⃗ .
Derivation of triangle law of vector addition
From triangle OCB,
OB2=OC2+BC2
OB2=(OA+AC)2+BC2 (eq.1)
In triangle ACB with ϴ as the angle between P and Q
cosΘ=ACAB
AC=ABcosΘ=QcosΘ
sinΘ=BCAB
BC=ABsinΘ=QsinΘ
Substituting the values of AC and BC in (eqn.1), we get
R2=(P+QcosΘ)2+(QsinΘ)2
R2=P2+2PQcosΘ+Q2cos2Θ+Q2sin2Θ
R2=P2+2PQcosΘ+Q2
therefore, R=P2+2PQcosΘ+Q2−−−−−−−−−−−−−−−−−√
Above equation is the magnitude of the resultant vector.
To determine the direction of the resultant vector, let ɸ be the angle between the resultant vector R and P.
From triangle OBC,
tanϕ=BCOC=BCOA+AC
tanϕ=QsinΘP+QcosΘ
therefore, ϕ=tan−1(QsinΘP+QcosΘ)
Above equation in the direction of the resultant vector.