Fundamental Theorem of Vectors

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