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ContentVector Components
Vector Components![]() 𝒒. Or in form of a scaled vector by simple geometry operation, that is a scalar 𝑘 times a directional vector 𝒅or the magnitude |𝒒| times a unit directional vector 𝒅. In other words, the study of vector components can only focus on the directional vector irrespective of its magnitude. Components of VectorIn general, the values that are used to represent a vector are called the components of the vector, and the number of components used to define a vector is equal to the number of dimensions of interest.One Dimension![]() 𝒅=(𝑑) Two Dimension![]() 𝒅=(𝑥,𝑦). ![]() 𝒅is a unit vector, the components of the vector can be expressed in terms of angles between the vector and coordinate axes, that is 𝑥= 𝒅=( Since the components of a vector is an ordered set, the rectangular Cartesian coordinate system of components 𝑥 and 𝑦 must be arranged in correct order following the right-hand rule. Three Dimension![]() 𝒅=(𝑥,𝑦,𝑧). ![]() 𝒅is a unit vector, the components of the vector can be expressed in terms of angles between the vector and coordinate axes, that is 𝑥= 𝒅=( Since the components of a vector is an ordered set, the rectangular Cartesian coordinate system of components 𝑥, 𝑦, 𝑧 and must be arranged in correct order following the right-hand rule. Vector in Space![]() 𝒂with initial point 𝐴 and terminal point 𝐵 can be expressed in term of coordinates 𝐴(𝑥1,𝑦1,𝑧1) and 𝐵(𝑥2,𝑦2,𝑧2). The vector 𝒂can also be interpreted as a displacement vector 𝐴𝐵displaced from point 𝐴 to point 𝐵. And the geometry of the directed line segment corresponding to the displacement vector can be specified by the numbers 𝑎1=𝑥2-𝑥1, 𝑎2=𝑦2-𝑦1, and 𝑎3=𝑧2-𝑧1 with respect to point 𝐴. These numbers are called components of the vector 𝒂with respect to the corresponding Cartesian coordinate system because vector 𝒂can be represented by these components. That is
And the length or the magnitude |𝒂| of vector 𝒂can be determined by the Pythagorean theorem geometrically. ![]() 𝒂are derived from the end points of the vector by subtracting the coordinates of initial point from the coordinates of the terminal point, the components of vector are independent of the choice of the initial point of the vector and are dependent on the magnitude and direction of the vector only. In other words, the components of vector 𝒂is a free vector bounded to point 𝐴 with respect to the corresponding Cartesian coordinate system. ![]() 𝒂and 𝒃are equal if and only if the corresponding component of two vectors are equal. That is 𝑎1=𝑏1, 𝑎2=𝑏2, and 𝑎3=𝑏3. As both the magnitude and the direction of the directed line segment of the corresponding vector can be obtained from the components of the corresponding vector, the geometry of the directed line segment in three dimensional space can also be determined by the ordered triple components of the vector in one-to-one relation with respect to the corresponding Cartesian coordinate system. However, the vector representation of quantity with magnitude and direction is always dependent on the choice of coordinate system. ©sideway ID: 191201202 Last Updated: 12/12/2019 Revision: 0 Ref: ![]() References
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