A Short Note on Components of a Vector

The components of a vector aid in the division of a given vector into parts that are oriented in different directions. It is sometimes necessary to divide a vector into its constituent parts in order to facilitate the execution of numerous arithmetic operations involving vectors. When a vector has components, each component represents a portion of the vector with respect to each of the axes of the coordinate system. When a vector is in a three-dimensional geometric plane, it is possible to compute the components of a vector as well.

The split of a vector is determined by the components of the vector. Each axis of the vector is divided, and we can compute the components of a vector using the information from the vector. The individual components of a vector can be combined later to form the complete vector representation. Generally speaking, vectors are represented in a two-dimensional coordinate plane with an x-axis and y-axis, or in a three-dimensional space containing the x-axis, the y-axis, and the z-axis respectively. Vectors are general mathematical representations that have both a direction and an amount of information.

It is the angle formed by the vector with respect to the positive x-axis in a two-dimensional coordinate system that determines the direction of the vector. Let V represent the vector, and represent the angle formed by the vector with respect to the positive x-axis. Furthermore, the components of this vector along the x and y axes are denoted by the letters Vx and Vy, respectively. The following expressions can be used to calculate the components of the equation.

      Vx= v. Cosθ and Vy = v. Sinθ

     |v|= √vx2 + vy2

The vectors are also represented in three-dimensional space by the equation A→ = a i∧ + b j∧ + c k∧ in the case of the vectors. The unit vectors along the x-axis, y-axis, and z-axis are represented by the letters I∧  j∧, and k∧, respectively. These unit vectors aid in the identification of the components of the vectors in relation to each of the axes. The components of vector A with respect to the x-axis, the y-axis, and the z-axis are denoted by the letters a, b, and c, respectively.

How to Find the Components of the Vector

The component form of the vector A→ is present in the equation below. The values a, b, and c are referred to as the scalar components of vector A→, and the values ai∧+ bj∧, and ck∧ are referred to as the vector components. In this case, the letters a, b, and c are also referred to as rectangular components. The magnitude of A is equal to the square root of the sum of the squares of its individual components.

|A| = √a²+ b² + c². 

Algebraic Components Using Components of a Vector

It is very simple to perform the various algebraic operations on vectors when the various components of the vector are taken into consideration. Think about two vectors, A→ = a₁I∧ + b1j∧ + c1k∧ and B→ = a₂I∧ + b2j∧ + c2k∧, and consider the relationship between them.

• Following is the equation for the addition of two vectors: A→ + B→ = (a1 + a2)i∧ + (b1+b2)j∧+ (c1+c2)k∧.
• In order to compute the subtraction of two vectors A→ and B→, we use the formula: A→ ₋ B→ = ( a₁ – a2)i∧ + (b1 – b2)j∧+ (c1-c2)k∧.
• The two vectors A→ and B→ are equal if and only if the following conditions are met: a₁ = a₂, b1 = b2, c1= c2.
• The multiplication of a vector by a scalar λ gives the following result: λA→ = λA→ = λa1I∧ + λb1j∧ + λc1k∧.

Conclusion

The x-component and the y-component of a vector in a two-dimensional coordinate system are commonly referred to as the vector’s components. The components of a vector aid in the division of a given vector into parts that are oriented in different directions. When a vector has components, each component represents a portion of the vector with respect to each of the axes of the coordinate system.

The individual components of a vector can be combined later to form the complete vector representation. Vectors are general mathematical representations that have both a direction and an amount of information.

The following expressions can be used to calculate the components of the equation.

        Vx= v. Cosθ and Vy = v. Sinθ

The component form of the vector A→ is present in the equation below. The values a, b, and c are referred to as the scalar components of vector A→, and the values ai∧+ bj∧, and ck∧ are referred to as the vector components.

The magnitude of A is equal to the square root of the sum of the squares of its individual components.

|A| = √a2+ b2 + c2.