Vector Subtraction

Vector subtraction of two vectors a and b is represented by a – b and it is nothing but adding the negative of vector b to the vector a. i.e., a – b = a + (-b).

Thus, the subtraction of vectors involves the addition of vectors and the negative of a vector. The result of vector subtraction is again a vector. The following are the rules for subtracting vectors:

• It should be performed between two vectors only (not between one vector and one scalar).
• Both vectors in the subtraction should represent the same physical quantity.

Vector Subtraction Formula

Here are multiple ways of subtracting vectors:

• To subtract two vectors a and b graphically (i.e., to find a – b), just make them coinitial first and then draw a vector from the tip of b to the tip of a.
• We can add –b (the negative of vector b which is obtained by multiplying b with -1) to a to perform the vector subtraction a – b. i.e., a – b = a + (-b).
• If the vectors are in the component form we can just subtract their respective components in the order of subtraction of vectors.

Thus, the addition formula can be applied as:

[Tex]\bold{\vec{A} – \vec{B} = \sqrt{A^2~+~B^2~-~2AB~cos~\theta}}[/Tex]

Note: [Tex]-\vec{B} [/Tex]is nothing but [Tex]\vec B [/Tex]reversed in direction.

Properties of Vector Subtraction

There are various properties of vector subtraction, some of those properties are:

• Any vector subtracted from itself results in a zero vector. i.e., a – a = 0, for any vector a
• The subtraction of vectors is NOT commutative. i.e., a – b is not necessarily equal to b – a
• The vector subtraction is NOT associative. i.e., (a – b) – c does not need to be equal to a – (b – c)
• (a – b) · (a + b) = |a|² – |b|²
• (a – b) · (a – b) = |a – b|² = |a|² + |b|² – 2 a · b

Vector Addition in Mathematics is the fundamental operation of vector algebra that is used to find the sum of two vectors. Vectors are mathematical quantities that have magnitude and direction. A vector can be represented by a line with an arrow pointing towards its direction and its length represents the magnitude of the vector.

Vector addition is achieved by taking the vector in 3D or 2D and then arranging them such that, the head of one vector is arranged touching the tail of the other vector and now a third vector joins the tail of the first vector with the head of the other vector gives the sum of the vectors.

In this article, we will learn about, vector definition, vector addition, laws of vector addition, and others in detail.