Analysis of the Subtraction Vectors

An object with magnitude and direction is called a vector. This quantity is represented mathematically or geometrically. A vector consists of initial points and terminal points, represented by arrows. The arrow shows a direction equal to the quantity, and a length equal to the magnitude of that quantity. Vectors do not have positions, even though they have magnitude and direction. A scalar differs from a vector in that it has magnitude, but no direction. For example, acceleration, velocity, and displacement are vector quantities, while mass, time, and speed are scalars.

Notation of vector

In general, vectors are represented by bold lowercase letters, such as a, or by arrows over the letters. You can also visually identify vectors by highlighting their initial and terminal points with an arrow above them. The point where a vector originates is the tail, and the point at which it terminates is called the head. 

Subtraction of Vectors 

Similar to the addition of vectors, the subtraction of vectors consists of changing the sign and adding it to the other.

Laws of Vector Subtraction

Triangle law of vector subtraction

The subtraction of vectors may be accomplished by applying the well-known triangle law, called the head-to-tail approach. According to this rule, two vectors may be put together in a position where the head of the first vector meets the tail of the second vector or vice versa. As a consequence, by subtracting the tail of the first vector from the head of the second vector, we may get the difference vector. 

The subtraction of vectors using the triangle method can be with the following steps:

Suppose there are two vectors: p and q

Now, draw a line PQ representing p with P as the tail and Q  as the head. Draw another line ST representing q with S as the tail and T as the head. Now join the line PT with P as the tail and T as the head. The line TP represents the resultant difference of the vectors p and q.                         

The line TP represents p – q.

The magnitude of p – q is: 

√(p2 + q2 – 2pq cos θ)

Where,

p= magnitude of the vector

q= magnitude of the vector

θ= angle between p and q.

Parallelogram Law of Vector Subtraction 

We can learn the subtraction vectors through the Parallelogram method, and we can also try subtracting vectors graphically. According to this law, if two vectors are co-existent at a point and are depicted in magnitude and direction by the two sides of a parallelogram drawn from a point, their subtraction comes by the diagonal of the parallelogram, with magnitude as well as direction passing through the same point.

If two vectors, p and q, denote 2 adjacent sides of the parallelogram, both pointing outwards, the diagonal drawn through the junction of the two vectors represents the subtraction. Its magnitude is shown by the square of the diagonal of the parallelogram, equal to the difference of the square of the adjacent sides.

Important Points to Keep in Mind When Subtracting Vectors

Here are some points to be kept in mind while studying the subtraction of vectors:

• Vectors are expressed as a combination of direction and magnitude, and they are represented graphically by an arrow.
• As long as we have the components of a vector, we can figure out what the final vector will look like.
• The subtraction of vectors can be accomplished through the application of the well-known triangle law, which is also known as the head-to-tail method.

Conclusion

The subtraction of vectors is finding the resultant of several vectors acting on a body. In subtraction to vectors, the resultant vector is independent of the order of vectors. 

Here are some of the important points about subtracting vectors:

• Vectors are not subtracted algebraically but geometrically.
• When calculating the results of a vector, it behaves independently from that vector.
• You can only subtract vectors that represent the same quantity. For instance, you can subtract two vectors that represent acceleration. But not when one vector represents acceleration and the other represents force. 
• Vector subtraction is both associative and commutative in nature. 
• You cannot subtract vectors and scalars.