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Components of a Vector
Components of a vector help to split a given vector into parts with respect to different directions. Sometimes there is a need to split the vector into its components to help perform numerous arithmetic operations involving vectors. Components of a vector represent part of the vector with reference to each of the axes of the coordinate system. The components of a vector can also be computed for a vector in a threedimensional geometric plane.
Let us learn more about the components of a vector, how to find the components of a vector, and the various arithmetic operations involving components of a vector.
What Are the Components of a Vector?
The components of a vector gives a split of the vector. The vector is split with reference to each of the axes, and we can compute the components of a vector. The individual components of a vector can be later combined to get the entire vector representation. Vectors are general represented in a twodimensional coordinate plane, with an xaxis, yaxis, or threedimensional space, containing the xaxis, yaxis, zaxis respectively. Vectors are general mathematical representations with direction and magnitude.
In a twodimensional coordinate system, the direction of the vector is the angle made by the vector with the positive xaxis. Let V be the vector and θ is the angle made by the vector with the positive xaxis. Further, we have the components of this vector along the x and y axis as \(V_x\), and \(V_y\) respectively. These components can be computed using the following expressions.
\(V_x\) = V.Cosθ, and \(V_y\) = V.Sinθ
V = \(\sqrt{V_x^2 + V_y^2}\)
Further, the vectors are also represented as \(\overrightarrow A = a\hat i + b \hat j + c \hat k\) in the threedimensional space. Here \(\hat i\), \(\hat j\), \(\hat k\), are the unit vectors along the xaxis, yaxis, and zaxis respectively. These unit vectors help in identifying the components of the vectors with reference to each of the axes. The components of vector A with respect to the xaxis, yaxis, zaxis, are a, b, c respectively.
How to Find the Components of a Vector?
The vector \(\overrightarrow A\) in the below image is called the component form. The values a, b, c are called the scalar components of vector A, and a\(\hat i\), b\(\hat j\), c\(\hat k\), are called the vector components. Here a, b, c are also termed as rectangular components. The magnitude of A is equal to the https://www.cuemath.com/algebra/squaresandsquareroots/square root of the sum of the squares of its individual components.
A = \(\sqrt{a^2 + b^2 + c^2}\)
Algebraic Operations Using Components of a Vector
The various algebraic operations on vectors can be easily performed by using the the various components of the vector. Let us consider two vectors \(\overrightarrow A = a_1\hat i + b_1 \hat j + c_1 \hat k\), and \(\overrightarrow B = a_2\hat i + b_2 \hat j + c_2 \hat k\).
 For the addition of two vectors \(\overrightarrow A\) and \(\overrightarrow B\) we have: \(\overrightarrow A + \overrightarrow B = (a_1 + a_2)\hat i + (b_1 + b_2)\hat j + (c_1 + c_2)\hat k\).
 For the subtraction of two vectors \(\overrightarrow A\) and \(\overrightarrow B\) we have: \(\overrightarrow A  \overrightarrow B = (a_1  a_2)\hat i + (b_1  b_2)\hat j + (c_1  c_2)\hat k\).
 The two vectors \(\overrightarrow A\) and \(\overrightarrow B\) are equal if: \(a_1 = a_2\), \(b_1 = b_2\), \(c_1 = c_2\).
 The multiplication of a vector with a scalar λ gives: \(λ\overrightarrow A = λa_1\hat i + λb_1 \hat j + λc_1 \hat k\).
Related Topics
The following topics are helpful for a better understanding of the components of a vector.
Examples of Components of a Vector

Example 1: Find the x and y components of a vector having a magnitude of 12 and making an angle of 45 degrees with the positive xaxis.
Solution:
The given vector is V= 12, and it makes an angle θ = 45º.
The x component of the vector = \(V_x\) = VCosθ = 12.Cos45º = 12.(1/√2) = 6√2.
The y component of the vector = \(V_y\) = VSinθ = 12.Sin45º = 12.(1/√2) = 6√2.
Therefore, the x component and the y components of the vector are both equal to 6√2.

Example 2: Find the vector from the components of a vector, having the xcomponent of 5 units, ycomponent of 12 units, and zcomponent of 4 units respectively.
Solution:
X Component of the vector = a = 5
Y Component of th vector = b = 12
Z Component of the vector = c = 4
The required vector is \(\overrightarrow V = a\hat i + b \hat j + c \hat k\)
Hence \(\overrightarrow V = 5\hat i + 12 \hat j + 4 \hat k\).
Therefore the required vector is \(\overrightarrow V = 5\hat i + 12 \hat j + 4 \hat k\).
FAQs on Components of a Vector
What Are the Three Components of a Vector?
The three components of a vector are the components along the xaxis, yaxis, and zaxis respectively. For a vector \(\overrightarrow A = a\hat i + b \hat j + c \hat k\), a, b, c are called the scalar components of vector A, and a\(\hat i\), b\(\hat j\), c\(\hat k\), are called the vector components.
Are the Components of a Vector, Also a Vector?
The components of a vector are also vectors. The vector \(\overrightarrow A = a\hat i + b \hat j + c \hat k\), has a, b, c as its components along the xaxis, yaxis, and zaxis respectively. Since the components of the vector has a magnitude and argument, which is along the direction of the respective axes, these components are also vectors.
Are Components of Vectors a Scalar?
The components of a vector are not scalars. The components of a vector are also vectors and they have a magnitude and direction. The components of a vector are also defined with respect to one of the axes in the coordinate plane or in the threedimensional space.
How To Find the Angle Made by the Vector with the Xaxis, From the Components of a Vector?
The angle made by the vector V with the xaxis is the angle θ, and the tan of the angle is equal to the y component of the vector, divided by the x component of the vector. Hence θ = \(Tan^{1}\frac{V_y}{V_x}\).
How Do you FInd that the Vectors are Collinear Based on Components of a Vector?
The collinearity of two vectors can be proved, if one vector is obtained by multiplying another vector with a constant value. Also for two collinear vectors, the respective components of the two vectors are in proportion. Two vectors \(\overrightarrow A = a_1\hat i + b_1 \hat j + c_1 \hat k\), and \(\overrightarrow B = a_2\hat i + b_2 \hat j + c_2 \hat k\) are said to be collinear if \(\overrightarrow A\) = λ\(\overrightarrow B\), and also \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) = λ.
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