Vector Formula Sheet

Vectors are mathematical entities characterized by magnitude and direction, often represented graphically as arrows, and used to denote quantities such as displacement, velocity, force, and more in physics, engineering, and mathematics.

Neetesh Kumar | May 07, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition & It's Types
• 2. Representation
• 3. Types
• 4. Addition of Vectors
• 5. Position Vector
• 6. Section Formula
• 7. Vector Equation of a Line
• 8. Test of Collinearity of Three Points
• 9. Scalar Product of two vectors (Dot Product)
• 10. Vector Product of two Vectors (Cross Product)
• 11. Shortest Distance between two Lines
• 12. Scalar triple product or Box Product or Mixed Product
• 13. Vector Triple Product

1. Definition & It's Types

Physical quantities are broadly divided into two categories viz (a) Vector Quantities & (b) Scalar quantities.

(a) Vector Quantities:

Any quantity, such as velocity, momentum, or force, with both magnitude and direction and for which vector addition is defined and meaningful is treated as a vector quantity.

(b) Scalar Quantities:

A quantity, such as mass, length, time, density, or energy that has size or magnitude but does not involve the concept of direction is called a scalar quantity.

2. Representation:

Vectors are represented by directed straight-line segments
Magnitude of $\vec{a}$ = | $\vec{a}$ | = Length of line segment PQ
Direction of $\vec{a}$ = P to Q

3. Types:

(a) Zero Vector or Null Vector:

A vector of zero magnitude, i.e., which has the same initial & terminal point, is called a ZERO VECTOR. It is denoted by $\vec{O}$.

(b) Unit Vector:

A vector of unit magnitude in the direction of a vector $\vec{a}$ is called a unit vector along a $\vec{a}$ and is denoted by $\hat{a}$ symbolically $\hat{a}$ = $\frac{\vec{a}}{|\vec{a}|}.$

(c) Colliinear Vector:

Two vectors are said to be collinear if their supports are parallel and disregard their direction. Collinear vectors are called $\bold{Parallel \space vectors}.$ If they have the same direction, they are named $\bold{like \space vectors}$ otherwise, they are $\bold{unlike \space vectors.}$ Symbolically, two non-zero vectors $\vec{a}$ & $\vec{b}$ are collinear if and only if, $\vec{a}$ = K$\vec{b}$, where K $\in$ R

(d) Coplanar Vectors:

Several vectors are called coplanar if their supports are all parallel to the same plane. Note that “TWO VECTORS ARE ALWAYS COPLANAR.”

(e) Equality of two Vectors:

Two vectors are said to be equal if they have

• The same length
• The same or parallel supports
• The same sense.

(f) Free Vectors:

If a vector can be translated anywhere in space without changing its magnitude & direction, then such a vector is called a free vector. In other words, the initial point of a free vector can be taken anywhere in space, keeping its magnitude & direction the same.

(g) Localized vectors :

A vector of a given magnitude and direction is called a localized vector if its initial point is fixed in space. Unless & until stated, vectors are treated as free vectors.

4. Addition of vectors :

$\bold{(a)}$ It is possible to develop an Algebra of Vectors, which proves useful in studying Geometry, Mechanics, and other branches of Applied Mathematics.
$\bold{(i)}$ If two vectors $\vec{a}$ & $\vec{b}$ are represented by $\overrightarrow{OB}$ & $\overrightarrow{OB}$ then their sum $\vec{a}$ + $\vec{b}$ is a vector represented by $\overrightarrow{OC}$ where OC is the diagonal of the parallelogram OACB.
$\bold{(ii)}$ $\vec{a}$ + $\vec{b}$ = $\vec{b}$ + $\vec{a}$ (Commutative)
$\bold{(iii)}$ ($\vec{a}$ + $\vec{b}$) + $\vec{c}$ = $\vec{a}$ + ($\vec{b}$ + $\vec{c}$) (associativity)

$\bold{(b)}$ Multiplication of Vector by Scalars :
$\bold{(i)}$ m($\vec{a}$) = ($\vec{a}$)m = m$\vec{a}$
$\bold{(ii)}$ m(n$\vec{a}$) = n(m$\vec{a}$) = (mn)$\vec{a}$
$\bold{(iii)}$ (m+n)$\vec{a}$ = m$\vec{a}$ + n$\vec{a}$
$\bold{(iv)}$ m($\vec{a}$ + $\vec{b}$) = m$\vec{a}$ + n$\vec{b}$

5. Position Vector:

Let O be a fixed origin, then the position vector of a point P is the vector $\overrightarrow{OP}$ If $\vec{a}$ & $\vec{b}$ are position vectors of two-point A and B, then,
$\overrightarrow{AB}$ = $\vec{b}$ - $\vec{a}$ = (Position Vector of point B) - (Position Vector of point A).

6. Section Formula:

If $\vec{a}$ & $\vec{b}$ are the position vectors of two points A & B, then the position vector of a point that divides AB in the ratio m : n is given by: $\frac{n\vec{a}+m\vec{b}}{m+n}.$

7. Vector Equation of a Line:

Parametric vector equation of a line passing through two point A($\vec{a}$) & B($\vec{b}$) is given by, $\vec{r}$ = $\vec{a}$ + t($\vec{b}$ - $\vec{a}$) where t is a parameter. If the line passes through the point A($\vec{a}$) and is parallel to the vector $\vec{b}$ then its equation is $\vec{r}$ = $\vec{a}$ + t$\vec{b}$, where t is a parameter.

8. Test of Collinearity of Three Points:

$\bold{(a)}$ Three points A, B, C with position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ respectively are collinear, if & only if there exist scalars x, y, z not all zero simultaneously such that ; x$\vec{a}$ + y$\vec{b}$ + z$\vec{c}$ = 0, where x + y + z = 0

$\bold{(b)}$ Three points A, B, and C are collinear if any two vectors $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CA}$ are parallel.

9. Scalar Product of two vectors (Dot Product) :

$\bold{(a)}$ $\vec{a}.\vec{b}$ = $|\vec{a}|.|\vec{b}|.$Cos($\theta$) where ($0 \le \theta \le \pi$), $\theta$ is angle between $\vec{a}$ & $\vec{b}.$ If $\theta$ is acute, then $\vec{a}.\vec{b}$ > 0 and if $\theta$ is obtuse then $\vec{a}.\vec{b}$ < 0.

$\bold{(b)}$ $\vec{a}.\vec{a}$ = $|\vec{a}|^2,$ $\vec{a}.\vec{b}$ = $\vec{b}.\vec{a}$ (commutative), $\vec{a}.(\vec{b}+\vec{c})$ = $\vec{a}.\vec{b}$ + $\vec{a}.\vec{c}$ (Distributive)

$\bold{(c)}$ $\vec{a}.\vec{b}$ = 0 $\iff$ $\vec{a} \perp \vec{b};$ ($\vec{a},\vec{b} \ne 0$)

$\bold{(d)}$ $\hat{i}.\hat{i} = \hat{j}.\hat{j} = \hat{k}.\hat{k} = 1; \hat{i}.\hat{j} = \hat{j}.\hat{k} = \hat{k}.\hat{i} = 0$

$\bold{(e)}$ Projection of $\vec{a}$ on $\vec{b}$ = $\frac{\vec{a}.\vec{b}}{|\vec{b}|}$

$\bold{Note:}$

• The vector component of $\vec{a}$ along $\vec{b}$ i.e. $\vec{a}_1$ = $(\frac{\vec{a}.\vec{b}}{\vec{b}^2})\vec{b}$ and perpendicular to $\vec{b}$ i.e. $\vec{a}_2$ = $\vec{a}$ $- (\frac{\vec{a}.\vec{b}}{\vec{b}^2})\vec{b}$

• The angle $\phi$ between $\vec{a}$ and $\vec{b}$ is given by Cos$\phi$ = $\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$ where $0 \le \phi \le \pi$

• If $\vec{a}$ = $a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b}$ = $b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ then $\vec{a}.\vec{b}=a_1b_1 + a_2b_2 + a_3b_3$
  $|\vec{a}| = \sqrt{a_1^2+a_2^2+a_3^2}$ , $|\vec{b}| = \sqrt{b_1^2+b_2^2+b_3^2}$

• $-|\vec{a}| |\vec{b}| \le \vec{a}.\vec{b} \le |\vec{a}| |\vec{b}|$

• Any vector $\vec{a}$ can be written as, $\vec{a}$ = $(\vec{a}.\hat{i})\hat{i}$ + $(\vec{a}.\hat{j})\hat{j}$ + $(\vec{a}.\hat{k})\hat{k}$

• A vector in the direction of the bisector of the angle between the two vectors $\vec{a}$ and $\vec{b}$ is $\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}.$ Hence, the bisector of the angle between the two vectors $\vec{a}$ and $\vec{b}$ is $\lambda(\hat{a} + \hat{b})$, where $\lambda \in R^+.$ Bisector of the exterior angle between $\vec{a}$ and $\vec{b}$ is $\lambda(\hat{a} - \hat{b})$, $\lambda \in R^+.$

• $|\vec{a} \plusmn\vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \plusmn 2\vec{a}.\vec{b}$

• $|\vec{a} + \vec{b} + \vec{c}|^2= |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a})$

10. Vector Product of two Vectors (Cross Product) :

$\bold{(a)}$ If $\vec{a}$ and $\vec{b}$ are two vectors and $\theta$ is the angle between them, then $\vec{a} \times \vec{b}=|\vec{a}||\vec{b}|$Sin($\theta)\hat{n}$, where $\hat{n}$ is the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{a}$, $\vec{b}$ and $\vec{n}$ forms a right-handed screw system.

$\bold{(b)}$ Lagranges Identity : For any two vectors $\vec{a}$ and $\vec{b}$; $(\vec{a} \times \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 - (\vec{a}.\vec{b})^2 = \begin{vmatrix} \vec{a}.\vec{a} & \vec{a}.\vec{b} \\ \vec{a}.\vec{b} & \vec{b}.\vec{b} \end{vmatrix}$

$\bold{(c)}$ Formulation of vector product in terms of scalar product: The vector product $\vec{a} \times \vec{b}$ is the vectors $\vec{c}$, such that

• $|\vec{c}| = \sqrt{\vec{a}^2\vec{b}^2 - (\vec{a}.\vec{b})^2}$
• $\vec{c}.\vec{a}=0; \vec{c}.\vec{b}=0$ and
• $\vec{a}, \vec{b}, \vec{c}$ form a right-handed system

$\bold{(d)}$ $\vec{a} \times \vec{b} = 0 \iff \vec{a}$ and $\vec{b}$ are parallel (collinear) $(\vec{a} \ne 0, \vec{b} \ne 0)$ i.e. $\vec{a} = K\vec{b},$ where K is a Scalar.

• $\vec{a} \times \vec{b} \ne \vec{b} \times \vec{a}$ (not Commutative)
• $(m\vec{a}) \times \vec{b} = \vec{a} \times (m\vec{b}) = m(\vec{a} \times \vec{b})$, where m is scalar.
• $\vec{a} \times (\vec{b} + \vec{c}) = (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c})$ (distributive)
• $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$ and $\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$
  

$\bold{(e)}$ If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ then $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

$\bold{(f)}$ Geometrically $|\vec{a} \times \vec{b}|$ = area of the parallelogram whose two adjacent sides are represented by $\vec{a}$ and $\vec{b}$.

$\bold{(g)}$

• Unit vector perpendicular to the plane of $\vec{a}$ and $\vec{b}$ is $\hat{n} = \plusmn \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$
• A vector of magnitude ‘r’ & perpendicular to the plane of $\vec{a}$ and $\vec{b}$ is $\plusmn \frac{r(\vec{a} \times \vec{b})}{|\vec{a} \times \vec{b}|}$
• If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then Sin$\theta$ = $\frac{|\vec{a} \times \vec{b}|}{|\vec{a}|| \vec{b}|}$

$\bold{(h)}$ Vector area:

• If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are the position vectors of 3 points A, B and C then the vector area of triangle ABC = $\frac{1}{2}[(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a})].$ The points A, B and C are collinear if $(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a}) = 0$
• Area of any quadrilateral whose diagonal vectors are $\vec{d}_1$ and $\vec{d}_2$ is given by $\frac{1}{2}|\vec{d}_1 \times \vec{d}_2|$.
• Area of $\Delta$ with adjacent sides as $\vec{a}$ and $\vec{b}$ is $= \frac{1}{2}|\vec{a} \times \vec{b}|$

11. Shortest Distance between two Lines :

Lines that do not intersect & are also not parallel are called skew lines. In other words, the lines that are not coplanar are skew. For Skew lines, the direction of the shortest distance vector would be perpendicular to both the lines. The magnitude of the shortest distance vector would be equal to that of the projection of $\overrightarrow{AB}$ along the direction of the line of the shortest distance, $\overrightarrow{LM}$ is parallel to $\vec{p} \times \vec{q}$ i.e.,

$\overrightarrow{LM}$ = |Projection of $\overrightarrow{AB}$ on $\overrightarrow{LM}$| = |Projection of $\overrightarrow{AB}$ on $\vec{p} \times \vec{q}$ | = $|\frac{\overrightarrow{LM}.(\vec{p} \times \vec{q})}{\vec{p} \times \vec{q}}| = |\frac{(\vec{b} - \vec{a}).(\vec{p} \times \vec{q})}{|\vec{p} \times \vec{q}|}|$

$\bold{(a)}$ The two lines directed along $\vec{p}$ and $\vec{q}$ will intersect only of shortest distance = 0 i.e. $(\vec{b} - \vec{a}).(\vec{p} \times \vec{q})$ = 0 i.e. $(\vec{b} - \vec{a})$ lies in the plane containing $\vec{p}$ and $\vec{q} \Rightarrow$ $[(\vec{b} - \vec{a}) \space \space \vec{p} \space \space \vec{q}]$

$\bold{(b)}$ If two lines are given by $\vec{r_1} = \vec{a_1} + K_1\vec{b}$ and $\vec{r_2} = \vec{a_2} + K_2\vec{b}$ i.e. they are parallel then, d = $|\frac{\vec{b} \times (\vec{a_2} - \vec{a_1})}{|\vec{b}|}|$

12. Scalar triple product or Box Product or Mixed Product:

$\bold{(a)}$ The scalar triple product of three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ is defined as: $(\vec{a} \times \vec{b}).\vec{c} = |\vec{a}||\vec{b}||\vec{c}|$.Sin$\theta.$Cos$\phi$ Where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ and $\phi$ is the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$. It is also defined as $[\vec{a} \space \space \vec{b} \space \space \vec{c}]$ called as box product.

$\bold{(b)}$ In a scalar triple product the position of dot & cross can be interchanged i.e. $\vec{a}.(\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}).\vec{c}$ OR $[\vec{a} \space \space \vec{b} \space \space \vec{c}]$ = $[\vec{b} \space \space \vec{c} \space \space \vec{a}]$ = $[\vec{c} \space \space \vec{a} \space \space \vec{b}]$

$\bold{(c)}$ $\vec{a}.(\vec{b} \times \vec{c}) = -\vec{a}.(\vec{c} \times \vec{b})$ i.e. $[\vec{a} \space \space \vec{b} \space \space \vec{c}]$ = $-[\vec{a} \space \space \vec{c} \space \space \vec{b}]$

$\bold{(d)}$ If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are coplanar $\iff$ $[\vec{a} \space \space \vec{b} \space \space \vec{c}] = 0 \Rightarrow \vec{a}$, $\vec{b}$ and $\vec{c}$ are Linearly Independent.

$\bold{(e)}$ Scalar product of three vectors, two of which are equal or parallel is 0 i.e., $[\vec{a} \space \space \vec{b} \space \space \vec{c}] = 0$

$\bold{(f)}$ [i j k] = 1: $[K\vec{a} \space \space \vec{b} \space \space \vec{c}]$ = K$[\vec{a} \space \space \vec{b} \space \space \vec{c}]$; $[(\vec{a} + \vec{b}) \space \space \vec{c} \space \space \vec{d}]$ = $[\vec{a} \space \space \vec{c} \space \space \vec{d}]$ + $[\vec{b} \space \space \vec{c} \space \space \vec{d}]$

$\bold{(g)}$

• The Volume of the tetrahedron OABC with O as origin & the position vectors of A, B, and C being $\vec{a}$, $\vec{b}$ and $\vec{c}$ are given by V = $\frac{1}{6}$ $[\vec{a} \space \space \vec{b} \space \space \vec{c}]$
• Volume of parallelopiped whose co-terminus edges are $\vec{a}$, $\vec{b}$ and $\vec{c}$ is $[\vec{a} \space \space \vec{b} \space \space \vec{c}]$

$\bold{(g)}$ We can directly remember these results

• $[\vec{a} - \vec{b} \space \space \vec{b} - \vec{c} \space \space \vec{c} - \vec{a}]$ = 0
• $[\vec{a} + \vec{b} \space \space \vec{b} + \vec{c} \space \space \vec{c} + \vec{a}]$ = 2$[\vec{a} \space \space \vec{b} \space \space \vec{c}]$
• $[\vec{a} \space \space \vec{b} \space \space \vec{c}]^2$ = $[\vec{a} \times \vec{b} \space \space \vec{b} \times \vec{c} \space \space \vec{c} \times \vec{a}]$ = $\begin{vmatrix} \vec{a}.\vec{a} & \vec{a}.\vec{b} & \vec{a}.\vec{c} \\ \vec{b}.\vec{a} & \vec{b}.\vec{b} & \vec{b}.\vec{c} \\ \vec{c}.\vec{a} & \vec{c}.\vec{b} & \vec{c}.\vec{c} \end{vmatrix}$

13. Vector triple product :

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be any three vectors, then that expression $\vec{a} \times (\vec{b} \times \vec{c})$ is a vector and is called vector triple product.
$\bold{(a)}$ $\vec{a} \times (\vec{b} \times \vec{c})$ = $(\vec{a}.\vec{c})\vec{b} - (\vec{a}.\vec{b})\vec{c}$

$\bold{(b)}$ $(\vec{a} \times \vec{b}) \times \vec{c}$ = $(\vec{a}.\vec{c})\vec{b} - (\vec{b}.\vec{c})\vec{a}$

$\bold{(c)}$ $(\vec{a} \times \vec{b}) \times \vec{c} \ne$ $\vec{a} \times (\vec{b} \times \vec{c})$

14. Linear Combination, Dependence, and Independence of Vectors:

$\bold{Linear \space Combination \space of \space Vectors}$
Given a finite set of vectors $\vec{a}$, $\vec{b}$ and $\vec{c},......$ then the vector $\vec{r} = x\vec{a} + y\vec{b} + z\vec{c}+.....$ is called a linear combination of $\vec{a}$, $\vec{b}$, $\vec{c}, .....$ for any x, y, z ..... $\in$ R. We have the following results:
$\bold{(a)}$ If $\vec{x_1}, \vec{x_2}, ....\vec{x_n}$ are n non zero vectors, and $k_1, k_2, ...,k_n$ are n scalars and if the linear combination $k_1\vec{x_1} + k_2\vec{x_2} + ....+ k_n\vec{x_n} = 0 \Rightarrow k_1=0, k_2 = 0.....k_n=0$ then we say that vectors $\vec{x_1}, \vec{x_2}, ....\vec{x_n}$ are Linearly Independent Vectors.

$\color{red} \bold{Related \space Pages:}$
Operation on Vectors
Determinants Formula Sheet
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