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 three-dimensional 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 two-dimensional coordinate plane, with an x-axis, y-axis, or three-dimensional space, containing the x-axis, y-axis, z-axis respectively. Vectors are general mathematical representations with direction and magnitude.
In a two-dimensional coordinate system, the direction of the vector is the angle made by the vector with the positive x-axis. Let V be the vector and θ is the angle made by the vector with the positive x-axis. 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 three-dimensional space. Here \(\hat i\), \(\hat j\), \(\hat k\), are the unit vectors along the x-axis, y-axis, and z-axis 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 x-axis, y-axis, z-axis, 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/squares-and-square-roots/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.
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 x-axis, y-axis, and z-axis 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 x-axis, y-axis, and z-axis 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 three-dimensional space.
How To Find the Angle Made by the Vector with the X-axis, From the Components of a Vector?
The angle made by the vector V with the x-axis 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.