what is triangle law of vector addition? explain it
Answers
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:
Scalar 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
What is Triangle 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 formula
→R=→A+→B
To obtain →R which is the resultant of the sum of →A and →B with the same order of magnitude and direction as shown in the figure.
Explanation:
Do I hv to explain more
Explanation:
Triangle law
In most of the situations, we are involved with the addition of two vector quantities. Triangle law of vector addition is appropriate to deal with such situation.
Triangle law of vector addition
If two vectors are represented by two sides of a triangle in sequence, then third closing side of the triangle, in the opposite direction of the sequence, represents the sum (or resultant) of the two vectors in both magnitude and direction.
Here, the term “sequence” means that the vectors are placed such that tail of a vector begins at the arrow head of the vector placed before it.
The triangle law does not restrict where to start i.e. with which vector to start. Also, it does not put conditions with regard to any specific direction for the sequence of vectors, like clockwise or anti-clockwise, to be maintained. In figure (i), the law is applied starting with vector,b; whereas the law is applied starting with vector, a, in figure (ii). In either case, the resultant vector, c, is same in magnitude and direction.
This is an important result as it conveys that vector addition is commutative in nature i.e. the process of vector addition is independent of the order of addition. This characteristic of vector addition is known as “commutative” property of vector addition and is expressed mathematically as :
a+b=b+a
If three vectors are represented by three sides of a triangle in sequence, then resultant vector is zero. In order to prove this, let us consider any two vectors in sequence like AB and BC as shown in the figure. According to triangle law of vector addition, the resultant vector is represented by the third closing side in the opposite direction. It means that :
Vectors operate with other scalar or vector quantities in a particular manner. Unlike scalar algebraic operation, vector operation draws on graphical representation to incorporate directional aspect.
Vector addition is, however, limited to vectors only. We can not add a vector (a directional quantity) to a scalar (a non-directional quantity). Further, vector addition is dealt in three conceptually equivalent ways :
graphical methods
analytical methods
algebraic methods
In this module, we shall discuss first two methods. Third algebraic method will be discussed in a separate module titled Components of a vector
The resulting vector after addition is termed as sum or resultant vector. The resultant vector corresponds to the “resultant” or “net” effect of a physical quantities having directional attributes. The effect of a force system on a body, for example, is determined by the resultant force acting on it. The idea of resultant force, in this case, reflects that the resulting force (vector) has the same effect on the body as that of the forces (vectors), which are added.
Resultant force
It is important to emphasize here that vector rule of addition (graphical or algebraic) do not distinguish between vector types (whether displacement or acceleration vector). This means that the rule of vector addition is general for all vector types.
It should be clearly understood that though rule of vector addition is general, which is applicable to all vector types in same manner, but vectors being added should be like vectors only. It is expected also. The requirement is similar to scalar algebra where 2 plus 3 is always 5, but we need to add similar quantity like 2 meters plus 3 meters is 5 meters. But, we can not add, for example, distance and temperature.
Vector addition : graphical method
Let us examine the example of displacement of a person in two different directions. The two displacement vectors, perpendicular to each other, are added to give the “resultant” vector. In this case, the closing side of the right triangle represents the sum (i.e. resultant) of individual displacements AB and BC.
Displacement
AC=AB+BC
The method used to determine the sum in this particular case (in which, the closing side of the triangle represents the sum of the vectors in both magnitude and direction) forms the basic consideration for various rules dedicated to implement vector addition.
