Operations on Vectors

Vector operations are operations defined on vectors to get sum results:

• Addition of Vectors
• Subtraction of Vectors
• Scalar Multiplication
• Equality of Vectors

Let’s discuss these operations in detail:

Addition of Vectors

Vector Addition is a fundamental operation in vector mathematics that combines two or more vectors to produce a resultant vector. This operation is essential in physics, engineering, computer science, and many other fields.

Vectors can be added using various methods such as

• Graphical Method
• Triangle Law
• Parallelogram Law
• Polygon Law

Properties of Vector Addition

Some of the common properties of vector addition are:

Commutative Property: Vector addition is commutative. For example , for two vector a and b this property say a +b = b + a.

Associative property: Vector addition is Associative in nature. For example , for three vector a , b and c this property say (a +b)+c = a+(b + c ).

Distributive property: Vector addition is distributive over Scaler Multiplication. For example λ(a + b ) = λa + λb , where λ is a scalar quantity and a and b are two vectors .

Existence of Inverse For every vector a there exists a another vector b such that a + b =0 i.e. b = -a .

Existence of Identity : For every vector a there exists a another vector b such that a + b = a i.e. b = 0.

Note: It’s important to recognize that two vectors can be added only if they are of same nature.

Subtraction of Vectors

Two vectors can be subtracted to get a new resultant vector. Vector subtraction is just similar to vector addition.

Vector Subtraction can be seen as addition of first vector and the negative or second vector or we can say that for two vectors a and b , a – b = a + (-b). Vector subtraction do follows the laws of vector addition just like vector addition.

Properties of Vector Subtraction

Some of the common properties of vector subtraction are:

• Commutative Property: Vector subtraction isn’t commutative i.e. a – b ≠ b – a.
• Associative Property: Vector subtraction is Associative in nature. For example , for three vector a , b and c this property say (a – b) – c = a- ( b – c ).
• Distributive Property: Vector subtraction is distributive over Scaler Multiplication. For example λ(a – b ) = λa – λb , where λ is a scalar quantity and a and b are two vectors.
• Existence of Inverse: For every vector a there exists a another vector b such that a – b =0 i.e. b = a.
• Existence of Identity: For every vector a there exists a another vector b such that a – b = a i.e. b = 0.

Scalar Multiplication

Scalar multiplication is an operation in vector mathematics where a vector is multiplied by a scalar (a real number). This operation scales the magnitude of the vector without changing its direction (unless the scalar is negative, which also reverses the direction).

For a vector [Tex]\mathbf{v} = \langle v_x, v_y, v_z \rangle[/Tex] and a scalar k:

[Tex]k\mathbf{v} = k \langle v_x, v_y, v_z \rangle = \langle k v_x, k v_y, k v_z \rangle[/Tex]

Equality of Vector

Two vector are said to be equal if they have the same magnitude and direction.

This means that if you have two vectors, V and W then they are equal only if ∣v∣=∣w∣ and they point in the same direction. In other words, they represent the same physical quantity or geometric quantity in space. For example two vector X = ai + bj + ck , and Y = pi + qj + rj are equal iff a = p , b= q and c =r.

Vectors are one of the most important concepts in mathematics. Vectors are quantities that have both magnitude and direction. A vector quantity is represented by an arrow above its head. Vectors help us understand the behaviour of directional quantities in 2D and 3D planes. Vectors are also used for determining the position and change of position of points.

Every vector follows a certain set of rules, known as the properties of vectors. It is highly important to know these properties to have a strong command of vector algebra. In this article, we will see the definition of a vector, the properties of vectors, and the properties of vector products.