# 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|= $\sqrt{x^{2}+y^{2}+z^{2}}$

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