Geometric addition of vectors
Although there are many ways of combining vectors or adding and subtracting vectors , The simplest and most straight forward method of combining vectors is by graphical method or geometrical method.
In Geometric vector addition we mainly use following two laws of vector addition:
1> Law of triangle of vector addition :
The law of triangle of vector addition states:
If , the two sides of a triangle represents two given vectors in magnitude and direction in same order , then third side drawn in opposite sense represents their vector sum.
Let there be two vectors and and the angle between them is as shown in the picture below:
Then , To find their sum( ) first of all we reposition the two vectors such that the head of vector exactly coincides with the tail of vector and then draw a vector from the tail of the vector to head of the vector , The newly drawn vector represents the vector b sum of vectors “a” and “b” , as shown in the figure below:
2> Law of parallelogram of vector addition:
It states that if two adjacent sides of a parallelogram represents two given vectors in magnitude and direction , then the diagonal starting from the intersection of two vectors represent their sum.
The example of law of parallelogram of vector addition is given in following picture:
Properties of vector addition:
Vector addition have following properties:
1> Commutative law:
This law states that the order of addition does not matter in vector addition. or,
2> Associative law:
This law states , when more than two vectors are added we can group them in any order, as we add them. Or,
Subtraction of vectors:
To define the subtaction of vectors first we need to define the negative vector of a vector.
The negative vector of vector is denoted by vector and is a vector with the same magnitude as of vector But with exactly opposite direction.
Adding has the same effect as subtracting , so we use the following formula to subtract a vector from another:
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