Laws of Vector Addition

What is a Vector?

Let’s go through Law of vector addition pdf.The picture below shows a vector:

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A vector has magnitude (that is the size) and direction:

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The length of the line or the arrow given above shows its magnitude and the arrowhead points in the direction.

Now, we can add two vectors by simply joining them head-to-tail, refer the diagram given below for better understanding:

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And it doesn't matter in which order the two vectors are added, we get the same result anyway:

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Notation:

A vector can often be written in bold, like a or b.

A vector can also be written as the letters of its tail and head with an arrow above it, as shown on the right side 

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In this article, we will discuss about law of vector and subtraction,triangle law of vector addition,Mathematics law of vector addition and Law of vector addition pdf.

What Are The Properties of Vector Addition?

Addition of vectors satisfies two important properties.

1. The Commutative law states that the order of addition doesn't matter, that is : A+B is equal to B+A.

2 The Associative law, which states that the sum of three vectors does not depend on which pair of vectors is added first, that is: (A+B)+C=A+(B+C).


What is the Addition of Two Vectors?

In most general terms, it says you can add two vectors and the result will be a vector. 

As an example, let’s consider 

V=R2={(a,b)|a,b∈R} Then for vector(v1)=(x1 , y1), (v2)=(x2 , y2) we have 

v1+v2=(x1 + x2 , y1 + y2)


Mathematics Law of Vector Addition-Triangle Law of Vector Addition:

Let’s discuss the triangle law of vector addition in law of vector addition pdf .Suppose, we have two vectors namely A and B as shown.

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Now the method to add these two vectors is very simple, what we need to do is to simply place the head of one vector over the tail of the other vector as shown in the figure below.

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Now after this, we need to join the other end points of both the vectors together as shown below,

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The resultant of the given vectors (A and B)is given by a vector C which represents the sum of vectors A and B that is, C=A+B

From Law of vector addition pdf ,vector addition is commutative in nature i.e.

If C=A+B ; then we can write C=B+A

Similarly if we need to subtract both the vectors using the triangle law then we simply reverse the direction of any vector and then add it to another one as shown below.

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Now we can mathematically represent this as , C = A – B (as they are in opposite directions). This is the Triangle Law of Vector Addition.


Mathematics Law of Vector Addition-Parallelogram Law of Addition

The Mathematics law of vector addition named Parallelogram law of Addition generally states that the sum of the squares of the length of the four sides of a parallelogram is equal to the sum of the squares of the length of the two diagonals of the parallelogram. In Euclidean Geometry, it is necessary that the parallelogram must have equal opposite sides.

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If ABCD is a parallelogram, then AB is equal to DC and AD is equal to BC. Then according to the definition of the parallelogram law, it is stated as

2(AB)2 + 2 (BC)2 =  (AC)2 + (BD)2

In case, the parallelogram is a rectangle, then the law can be stated as,

2(AB)2 + 2 (BC)2=  (AC)2

Because in a rectangle, two diagonals are of equal lengths. i.e., (AC=BD)


Parallelogram Law of Addition of Vectors Procedure

The steps for the parallelogram law of the addition of vectors are given below:

Step 1) Draw a vector using a suitable scale in the direction of the vector

Step 2) In step 2, you need to draw the second vector using the same scale from the tail of the first given vector

Step 3) Bow, you need to treat these vectors as the adjacent sides and then complete the parallelogram

Step 4) Now, the diagonal formed basically represents the resultant vector in both magnitude and direction


What Are The Essential Conditions For The Addition of Any Vectors?

The essential condition for the addition of two vectors is simply that they should be like vectors, that is the vectors should have the same dimensions and the same units. For example, a Force vector with another force vector can be added, when they are expressed in the same units, but can not add force and velocity as they have different dimensions.

For example:If we have velocities of 30 meters/second and 50 meter/second in given directions we can add them easily but we can not directly add the velocities of 3km/Second and say 500 meters/second unless both are converted to the same units.

If the two vectors belong to the same vector space they have the same dimension,but it is also possible to add two vectors with different dimensions.

For example, we have a vector A=3i+4j and a vector B=8i+5j+9k then we can also find sum although they have different dimension.Here we have to consider A=3i+4j+0k. The sum of the vectors A+B = 11i+9j+9k

In simple words we can say that two vectors can be added if and only if they have the same unit.


Questions to be Solved :

Question 1) Given the vectors A = 2i + 6j - 3k and B = 3i - 3j + 2k. Find A+B.

Answer) Let’s add the given vectors,

 A = 2i + 6j - 3k + B = 3i - 3j + 2k = (2+3)i+(6-3)j+(-3+2)k

Therefore, A+B = 5i+ 3j-1k

FAQ (Frequently Asked Questions)

1. What Are The Laws Of Vectors?

The parallelogram law of vector addition states that "If any two vectors acting simultaneously at a point are represented both in direction and magnitude by two adjacent sides of a parallelogram drawn from the point, then the diagonal of parallelogram through that point of the parallelogram represents the resultant both in magnitude and direction."

2. State The Parallelogram Law Of Vector Addition?

According to the Parallelogram law of vector addition, if any two vectors a and b represent two sides of a parallelogram in magnitude and direction, then their sum a + b is equal to the diagonal of the parallelogram through their common point in magnitude and direction.

3. What is The Triangle Law Of Vector Addition?

The 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 triangle’s third side represents the direction and the magnitude of the resultant vector.

4. What is Meant By Vector Addition?

Vector addition can be defined as the operation of adding two or more vectors together into a vector sum. The parallelogram law gives the rule for vector addition of two or more vectors. For two vectors, the vector sum can be obtained by placing them head to tail and drawing the vector from the free tail to the free head.