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Fundamental Theorem of Vectors

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Anti-Parallel Vectors

Any quantity having magnitude as well as direction refers to vectors. Thus, vectors can be defined as the directed line segment x. Here, the arrow above x represents the direction of the vector. For instance, if AB is a vector that points A is called the initial point from where the vector starts and point B refers to the terminal point where a vector ends. The distance between these two points is known as the length or magnitude of \[\vec{x}\].


What Are Position Vectors?

The line drawn from a point to fixed origin having a direction in three-dimensional space represents a position vector. For instance, the position vector of a point P having coordinates (x, y, z) with respect to fixed origin O having coordinates (0, 0, 0) is represented by |OP|= \[\sqrt{x^{2}+y^{2}+z^{2}}\]

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The above diagram represents the position vector with a magnitude r and having a direction from fixed origin O to point P. 


Different Types of Vectors

  • Zero Vector: Any vector having the same initial and ending points refers to zero vector and is represented by \[\vec{0}\] . 

  • Unit Vector: Any vector having a magnitude equal to one refers to a unit vector. Thus, \[\hat{x}\] is a unit vector where 丨\[\hat{x}\]丨= 1.

  • Co-initial Vectors: Two vectors having the same initial points refers to co-initial vectors. 

  • Collinear Vectors: These are also called as parallel or antiparallel vectors having the same direction. 

  • Equal Vectors: Two vectors having the same magnitude, as well as direction, refers to equal vectors, represented as \[\vec{x}\] = \[\vec{y}\].

  • Parallel Vectors: The vectors having similar magnitude, but have opposite direction refers to parallel vectors.

  • Non-collinear Vectors: Two vectors acting in different directions are called independent or non-antiparallel vectors. 

  • Coplanar Vectors: Two planar vectors are always coplanar to each other. 


Some Fundamental Theorems of Vectors

  • Two Dimensions: Let \[\vec{p}\] and \[\vec{q}\] be two non-parallel or antiparallel vectors. Any vector \[\vec{x}\] lying in the plane of two non-collinear vectors can be represented as a linear combination of \[\vec{p}\] and \[\vec{q}\]. If L and M are two scalar quantities, then the equation becomes:

L Χ \[\vec{p}\] + M Χ \[\vec{q}\] = \[\vec{x}\].

  • Three Dimensions: Let \[\vec{p}\], \[\vec{q}\], and \[\vec{r}\] be three non-parallel or antiparallel vectors. Any vector \[\vec{x}\] lying in the plane of three non-collinear vectors can be represented as a linear combination of \[\vec{p}\], \[\vec{q}\], and \[\vec{r}\]. If L, M, and N are three scalar quantities, then the equation becomes:

L Χ \[\vec{p}\] + M Χ \[\vec{q}\] + N Χ \[\vec{r}\] = \[\vec{x}\].

  • Linear Independence: A system of vectors \[\vec{x_{1}}\], \[\vec{x_{2}}\],\[\vec{x_{3}}\]......,\[\vec{x_{n}}\] are linearly independent if for scalar quantities, say,\[a_{1}\], \[a_{2}\], ……., \[a_{n}\] = 0, if

\[\vec{p}\] = \[\vec{x_{1}}\]\[a_{1}\] + \[\vec{x_{2}}\]\[a_{2}\] + \[\vec{x_{3}}\]\[a_{3}\] + …….\[\vec{x_{n}}\]\[a_{n}\] = 0

  • Linear Dependence: A system of vectors \[\vec{x_{1}}\], \[\vec{x_{2}}\],\[\vec{x_{3}}\]......,\[\vec{x_{n}}\] are linear dependent if for scalar quantities, say, \[a_{1}\], \[a_{2}\], ……., \[a_{n}\] is not equal to 0, if

\[\vec{p}\] = \[\vec{x_{1}}\]\[a_{1}\] + \[\vec{x_{2}}\]\[a_{2}\] + \[\vec{x_{3}}\]\[a_{3}\] + …….\[\vec{x_{n}}\]\[a_{n}\] = 0


Theorem 1: 

If \[\vec{p}\] and \[\vec{q}\] be two non - anti-parallel vectors, then every vector \[\vec{x}\], which is coplanar with \[\vec{p}\] and \[\vec{q}\] can be expressed in unique combination expressed as:

a\[\vec{p}\] + b\[\vec{q}\] = \[\vec{x}\], where a and b are scalar quantities of the respective vectors.

If \[\vec{p}\] and \[\vec{q}\] are two vectors that are perpendicular to each other, then these vectors can be drawn along the X and Y-axis. If \[\hat{x}\] and \[\hat{y}\] be two unit vectors, then it can be represented by:

\[\vec{t}\] = a\[\vec{x}\] + b\[\vec{y}\]


Theorem 2: 

If \[\vec{p}\], \[\vec{q}\], and \[\vec{r}\] be three non- antiparallel vectors, then every vector \[\vec{x}\] can be uniquely expressed in linear combination expressed as:

a\[\vec{p}\] + b\[\vec{q}\] + c\[\vec{r}\] = \[\vec{x}\], where a, b, and c are scalar components of the respective vectors.


Theorem 3: 

If three vectors namely;

\[\vec{p}\] = \[p_{1}\]\[\hat{i}\] + \[p_{2}\]\[\hat{j}\] + \[p_{3}\]\[\hat{k}\], \[\vec{q}\] = \[q_{1}\]\[\hat{i}\] + \[q_{2}\]\[\hat{j}\] + \[q_{3}\]\[\hat{k}\] and \[\vec{r}\] = \[r_{1}\]\[\hat{i}\] + \[r_{2}\]\[\hat{j}\] + \[r_{3}\]\[\hat{k}\] are coplanar, then \[p_{1}\]\[p_{2}\]\[p_{3}\]\[q_{1}\]\[q_{2}\]\[q_{3}\]\[r_{1}\]\[r_{2}\]\[r_{3}\]


Important Points to Remember:

  • Three points with position vectors, say, \[\vec{p}\], \[\vec{q}\], and \[\vec{r}\] are said to be collinear or antiparallel vectors if and only if there exist scalars that are not equal to zero.

           a\[\vec{p}\] + b\[\vec{q}\] + c\[\vec{r}\] = 0 and a + b + c = 0

  • Four points with position vectors, say, \[\vec{p}\], \[\vec{q}\], \[\vec{r}\] and \[\vec{w}\] are said to be coplanar if and only if there exist scalars such that the sum of any two scalars is not equal to zero.

            a\[\vec{p}\] + b\[\vec{q}\] + c\[\vec{r}\] + d\[\vec{s}\] = 0 and a + b + c + d= 0

FAQ (Frequently Asked Questions)

1. Define Vector Methods.

Consider the scalar quantities called temperature and mass. These quantities have the magnitude only and no direction. However, the vectors are defined with magnitude having direction. In vectors, a line of a defined length will be pointing towards some direction, and thus it defines the vector methods. To define the magnitude of a vector A, it is always defined by | A | or simply as A. The resultant direction of two antiparallel vectors is defined by the vector having greater magnitude.

2. What is the Angle Formed by the Antiparallel Vectors?

Anti-parallel vectors are defined by the vectors moving in the opposite direction. Thus, it defines the angle between two vectors as 180 degrees. You can visualize it by taking any vector pointing to any random direction. Now you can rotate it with 180 degrees. Hence,  the final position and direction of the vector will be antiparallel to the original one. Thus the angle formed between both of them is 180. Also, these are defined as having exactly the opposite direction.