Subtraction of Vectors

It is possible to exert mathematical operations on the vectors to a certain extent. Some of these operations are addition, subtraction and multiplication. The division of the vectors is not supposed to be of much significance. This is so because the division of vectors can be defined as a function of inverse multiplication of the given vectors. Similarly, the subtraction of vectors can also be seen as a function of the addition of the vectors. Just, in this case, we need to make one of the given vectors a negative. 

What is the subtraction of vectors?

Vector subtraction is the subtraction of two vectors. In this, the negative of one vector is added to the other vector that we were provided with. Therefore, the subtraction of vectors can also be defined as the addition of one of the vectors with the negative of the other given vector. It is to be noted that the result that we will get after subtracting two vectors will also be a vector. 

The property of subtraction of vectors can only be applied to the entities that are vectors themselves. This means that the quantities that we are subtracting, both of those quantities need to be vectors. If even one of the given quantities is a  scalar, then the formula for subtraction of vectors will not apply to it. 

Also, the vectors that are being subtracted need to represent the same physical quantity that they belong to. If their physical quantities or units are different, then those two vectors will not be possible to be subtracted. Therefore, the subtraction for those vectors will not be possible. This principle is quite similar to the addition of vectors. Both the given vectors need to possess the same units so that they can give their resultant that carries the same unit. 

What is the formula for subtraction of vectors?

The subtraction of two given vectors is simply represented as:

Using the triangle law of vector subtraction;

The difference between two vectors will give the resultant of the two given vectors. 

Resultant =

Similarly, for the parallelogram law of vector subtraction;

If the given vectors are acting in two different directions, then the resultant of their difference will be given in the following way:

Resultant =

The case will remain the same for the vector addition if the resultant of the vector does not lie in between the given two vectors. 

Properties of vector subtraction

The subtraction of vectors stands true for this algebraic property. Considering that both ‘a’ and ‘b’ have the same units, expanding them by the binomial expansion will lead to the same result. 

This can also be defined as the product of two results of the subtraction of vectors. When the binomial expansion theorem is applied to these two, the outcome that we will gain will be the same. 

• The subtraction of vectors is not an associative property. 

Both the problems on the left-hand side as well as the right-hand side will result in two different results. Hence, we can say that the property of being associative is not held during the calculations of the subtraction of vectors. 

• The subtraction of two vectors is not a commutative property.

The problems on both sides of the ‘not equal to’ sign are two different quantities. The outcomes that we will get out of those two calculations will not be the same. Hence, we can say that the subtraction of vectors is not commutative. 

• The subtraction of one vector from itself will result in the zero vector. 

This result will be held by any vector ‘a^→ ‘.

Conclusion

The subtraction of two vectors can be defined as the function of the addition of two vectors. The methods that are used to find the subtraction of vectors are the same as the methods that are used to find the sum of vectors. These methods are the triangle law of vectors and the parallelogram law of vectors. The vectors do not follow the properties of commutativity and associativity in the case of subtraction. And, for vector subtraction, both the entities are needed to be vectors.