Parallelogram Vector addition

One of the most common vector operations that is frequently encountered is addition operation. Addition of two or more vectors to arrive to a vector sum is called vector addition. Consider two vectors,  u=(u1, u2) and  v=( v1, v2), the sum of these vectors would be vector u + vector v= (u1+v1, u2+v2). This sum is called the resultant vector. There are various methods to find the resultant vector namely, parallelogram method, component method, graphical method, cosine method, polygon method etc.  

Parallelogram Vector addition:  In this method first two vectors are drawn such that their initial points coincide. Then the other two lines are drawn to form a parallelogram. The resultant would be the diagonal of the parallelogram drawn from the initial point to the opposite vertex of the parallelogram.

Vector addition component method is one way used in adding vectors. Component means ‘part ’and hence they can be considered as the coordinates of the point that is associated with the vector. In a Euclidean plane consider two vectors, u=(u1, u2) and v=( v1, v2), the resultant vector which is the sum of these vectors is given by, u+v = (u1+v1, u2+v2). In a three dimensional space, given vectors u=(u1, u2, u3) and v=( v1, v2,v3) the method would be similar to the method used in addition of vectors in a Euclidean plane. So, u+v = (u1+v1, u2+v2, u3+v3). We can finally conclude that vector-addition is just like the normal addition, component by component.

Let us now learn the vector addition graphical method, consider two vectors, u=(4,3) and v=(1,4) in the plane. Using the component method of vector-addition the sum can be given as, u+v = (4+1, 3+4) = (5,7). Using the graphical method we get the same resultant vector by taking one vector whose direction and magnitude is unchanged and placing its end at the unchanged vector’s tip, and joining the origin and the new location of the displaced vector using an arrow. This procedure in general works for addition of vectors. For any two given vectors u and v in the plane, the sum of the vectors in general can be graphically represented as the vector addition diagram given below

As the vectors in a two dimensional space lie in the same plane, any two vectors in a three dimensional space also lie in the same plane and hence graphical method works well for vector-addition in a 3-dimensional space.

Understanding the Concept of Linear Combination

In mathematics the equations are very important. The problems can be converted into equations and then the equations can be solved with the help of multiple methods. The solutions obtained must be checked for their feasibility. This is because all the solutions obtained will not be feasible. Only the solutions that are feasible must be selected, otherwise the answer might go wrong. One must be very careful in selecting the solutions of the equations.

In case of linearity the degree of the given equation is ‘1’. The linear combination can be formed with the help of an equation. There need to be constants for this process to be performed. It also involves the simple process of addition. Basically it is the formation of an expression. This expression can be formed with the help of constants and the simple addition process.

The term in the expression must be multiplied with a constant and the answers obtained must be added. This will give the required solution. The linear combinations can be very helpful and can have various applications in mathematics.

The process of solving linear combinations must be learnt thoroughly to appreciate the concept. The examples will help in better explaining the concept. This concept is related to the concept of linear algebra. It is a branch of algebra and has to be learnt to understand this concept. This concept is also very helpful in vector theory and is a part of vector theory.

Vectors are different from scalar quantities. Both are quantities which have magnitude. The difference between vector and scalar quantities is that the vector quantities have a direction attached to them. But the scalar quantities do not have a direction attached to them.

This is the basic difference between the two quantities. So, one has to be very careful in dealing with these types of quantities. The direction can play a very important role. If the direction is not properly denoted the answer can go wrong. So, in the case of vector quantities, the direction plays a very important role. But in case of scalar quantities the magnitude plays the most important role.

The change in magnitude can make the whole answer go wrong in the case of scalar quantities. But in the case of vector quantities both magnitude and direction are important. Even if one goes wrong the whole answer goes wrong.