Vector

 Scalars and Vectors

Scalar:

A physical quantity that is completely described by its magnitude is a scalar.

$\to$ Examples: mass, distance, speed etc.

Vector:

A physical quantity having both magnitude and direction is a vector.

$\to$ Examples: weight, displacement, velocity, momentum etc.

Elementary area, elementary volume, elementary length, elementary current, etc. can be regarded as vectors.

A quantity having both magnitude and direction may not be a vector. To be a vector, it should follow the laws of vector addition as well.

$\to$ Examples: time, surface tension, electric current, etc.

All potentials, all fluxes, and energy are scalars.

$\to$ Examples: Electric Potential, Magnetic Potential, Gravitational potential, electric flux, magnetic flux, kinetic energy, magnetic energy etc.

All flux densities, scalar gradients, field strengths/ intensities, and dipole moments are vectors.

$\to$ Examples: Electric flux density, temperature gradient, gravitational field intensity, etc.

Polar vectors: Vectors associated with linear directional effect

$\to$ Examples: Force, momentum, velocity, displacement acceleration etc.

Axial vectors: Vectors associated with rotation about an axis.

$\to$ angular displacement, angular velocity angular acceleration, moment of force, angular momentum, etc.

Unit vector: A vector with unit magnitude is unit vector.

$\hat{a} = \dfrac{\overrightarrow{a}}{|\overrightarrow{a}|}$

$\to$ A unit vector is unitless and dimensionless. It possesses the only direction.

Null vector: A vector with zero magnitude is null vector.

$\to$ It has arbitrary direction

Laws of vectors

Triangle Law of Vector Addition
when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle taken in reverse order represents the magnitude and direction of the resultant vector.
$\overrightarrow{R} = \overrightarrow{AB} + \overrightarrow{BC}$

Parallelogram law of Vector Addition

when two vectors are represented as two adjacent sides of the parallelogram both pointing away from the common vertex, then the resultant is represented by the diagonal of the parallelogram passing through the same common vertex.

Polygon law of vector addition

It states that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

Lami's Theorem It states that, if three coplanar vectors $\overrightarrow{P}, \overrightarrow{Q} and \overrightarrow{R}$ act at a point, then, their magnitude is proportional to the sine of angle between other two vectors.

$\dfrac{P}{\sin \alpha} = \dfrac{Q}{\sin \beta} = \dfrac{R}{\sin \gamma}$

Algebra of Vectors

Addition of Vectors

If $\overrightarrow{A}$ and $\overrightarrow{B}$ be any two vectors inclined at angle $\theta$. Then, $\overrightarrow{R}$ is the resultant of two vectors which makes an angle $\alpha$ with angle $\overrightarrow{A}.$

Magnitude and direction of $\overrightarrow{R}$

$\to$ Magnitude of $\overrightarrow{R} = R = \sqrt{A^2 + 2AB \cos \theta + B^2}$

$\to$ Direction of $\overrightarrow{R}$ , $\alpha= \dfrac{B \sin \theta}{A + B \cos \theta}$

Cases

$\to$ If $\theta = 0^{\circ}, R = A + B, \alpha =0^{\circ}$

$\to$ If $\theta = 90^{\circ}, R = \sqrt{A^2 + B^2}, \alpha =\tan^{-1} \Bigg(\dfrac{B}{A}\Bigg)$

$\to$ If $\theta = 180^{\circ}, R = |A - B|, \alpha = 0^{\circ}$ if $A > B$ and $\alpha = 180^{\circ}$ if $B>A$

$\to$ If $A = B$, $R = 2A \cos \dfrac{A}{2}$

$\to$ Limits $A-B \leq |\overrightarrow{A} + \overrightarrow{B}| \leq A + B$

Difference of Vectors

The difference of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is denoted by $\overrightarrow{R} =\overrightarrow{A}-\overrightarrow{B}$. $\overrightarrow{R}$ is the difference of two vectors which makes an angle $\alpha$ with angle $\overrightarrow{A}.$

Magnitude and direction of $\overrightarrow{R}$

$\to$ Magnitude of $\overrightarrow{R} = R = \sqrt{A^2 - 2AB \cos \theta + B^2}$

$\to$ Direction of $\overrightarrow{R}$ , $\alpha= \dfrac{B \sin \theta}{A - B \cos \theta}$

Cases

$\to$ If $\theta = 0^{\circ}, R = A - B, \alpha = 0^{\circ}$ if $A > B$ and $\alpha = 180^{\circ}$ if $B>A$

$\to$ If $\theta = 90^{\circ}, R = \sqrt{A^2 + B^2}, \alpha =\tan^{-1} \Bigg(\dfrac{B}{A}\Bigg)$

$\to$ If $\theta = 180^{\circ}, R = A +B, \alpha = 0^{\circ}$

$\to$ If $A = B$, $R = 2A \sin \dfrac{A}{2}$

$\to$ Limits $A-B \leq |\overrightarrow{A} - \overrightarrow{B}| \leq A + B$

Resolution of Vectors

$\to$ A vector can be resolved into any number of components. But, it has 2 rectangular components in plane and 3 rectangular components in space.

$\to$ The component of the vector is a vector.

$\to$ The rectangular component of the vector can never be greater than the vector itself.

For $\overrightarrow{A} = A_x \hat{i} + A_y \hat{j}$

Horizontal Component: $A_x = A \cos \theta$

Vertical Component: $A_y = A \sin \theta$

$\to$ If the components are given then

$A = \sqrt{{A_x^2+ A_y^2}}$

$\alpha = \tan^{-1} \Bigg(\dfrac{A_y}{A_x}\Bigg)$ where $\alpha$ is angle with horizontal.

Product of Vectors

Scalar Product

The scalar product of $\overrightarrow{A}$ and $\overrightarrow{B}$ inclined at angle $\theta$ is given by:

$\overrightarrow{A}. \overrightarrow{B} = AB \cos \theta$

$\to$ Scalar product of two vectors is the product of vector and projection of second vector in it.

$\to$ when $\theta =0^{\circ}, \overrightarrow{A}. \overrightarrow{B} = AB (\text{maximum})$

$\to$ when $\theta =90^{\circ} \overrightarrow{A}. \overrightarrow{B} = 0 (\text{Perpendicular condition})$

$\to$ when $\theta =180^{\circ}, \overrightarrow{A}. \overrightarrow{B} = -AB (\text{minimum})$

$\to$ Limit: $-AB \leq \overrightarrow{A}. \overrightarrow{B} \leq AB$

$\to$ Scalar product of vectors results in scalar.

$\to$ Scalar product is commutative i.e. $\overrightarrow{A}. \overrightarrow{B} = \overrightarrow{B}. \overrightarrow{A}$

$\to \hat{i}. \hat{i} = \hat{j}.\hat{j} = \hat{k}.\hat{k} = 1$

$\to \hat{i}. \hat{j} = \hat{j}.\hat{k} = \hat{k}.\hat{i} = 0$

$\to$ If components are given:

$\overrightarrow{A}. \overrightarrow{B} = (A_x \hat{i} + A_y \hat{j}).(B_x \hat{i} + B_y \hat{j}) = A_xB_x + A_yB_y$

$\to$ Examples: $\phi = \overrightarrow{B}. \overrightarrow{A}, W = \overrightarrow{F}. \overrightarrow{s}$

Vector Product

The vector product or cross product of $\overrightarrow{A}$ and $\overrightarrow{B}$ inclined at angle $\theta$ is given by:

$\overrightarrow{A} \times \overrightarrow{B} = AB \sin \theta \hat{n}$

$\hat{n}$ is the unit vector perpendicular to the plane containing $\overrightarrow{A}$ and $\overrightarrow{B}$

$\to$ Magnitude of vector product of two vectors gives the area of parallelogram which has two vectors as adjacent vectors.

$\to$ when $\theta =0^{\circ}, \overrightarrow{A} \times \overrightarrow{B} = 0$

$\to$ when $\theta =90^{\circ} \overrightarrow{A} \times \overrightarrow{B} = 1$

$\to$ Limit: $0 \leq \overrightarrow{A} \times \overrightarrow{B} \leq AB$

$\to$ Vector product of vectors results in vector.

$\to$ Vector product is not commutative i.e. $\overrightarrow{A} \times \overrightarrow{B} \neq \overrightarrow{B} \times \overrightarrow{A}$

$\to \overrightarrow{A} \times \overrightarrow{B} =- \overrightarrow{B} \times \overrightarrow{A}$

$\to \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$

$\to \hat{i} \times \hat{j} = \hat{j} \times \hat{k} = \hat{k} \times \hat{i} = 1$

$\to \hat{j} \times \hat{i} = \hat{k} \times \hat{j} = \hat{i} \times \hat{k} = 1$

$\to$ If components are given:

$\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$

$\to$ Examples: $\overrightarrow{\tau} = \overrightarrow{r} \times \overrightarrow{F}, \overrightarrow{v} = \overrightarrow{\omega} \times \overrightarrow{r}$

Projection of Vectors

The projection of $\overrightarrow{B}$ on $\overrightarrow{A}$ is $B \cos \theta$

We know,

$\overrightarrow{A}. \overrightarrow{B} = AB \cos \theta$

$\dfrac{\overrightarrow{A}. \overrightarrow{B}}{A} = B \cos \theta$

$\hat{A}. \overrightarrow{B} = B \cos \theta$

Hence,

$\to$ The projection of $\overrightarrow{B}$ on $\overrightarrow{A}$ is $\hat{A}. \overrightarrow{B}$

$\to$ The projection of $\overrightarrow{A}$ on $\overrightarrow{B}$ is $\hat{B}. \overrightarrow{A}$

Better You Know

$\overrightarrow{a}$ and $\overrightarrow{b}$ be any two vectors then

$\to$ when $\theta =0, \overrightarrow{a}+\overrightarrow{b}= a+b$

$\to$ when $\theta =90^{\circ}, \overrightarrow{a}+\overrightarrow{b}= \sqrt{a^2+b^2}$

$\to$ when $\theta =180^{\circ}, \overrightarrow{a}+\overrightarrow{b}= a-b$

$\to$ when $| \overrightarrow{a}|=| \overrightarrow{b}|=A, \overrightarrow{a}+\overrightarrow{b}= 2 A \cos \dfrac{\theta}{2}$

$\to$ when $| \overrightarrow{a}|=| \overrightarrow{b}|=A, \overrightarrow{a}-\overrightarrow{b}= 2 A \sin \dfrac{\theta}{2}$

$\to$ when $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{a} + \overrightarrow{b}|, \text{then} \theta = 120^{\circ}$

$\to$ when $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{a}- \overrightarrow{b}|, \text{then} \theta = 60^{\circ}$

$\to$ when $| \overrightarrow{a}+\overrightarrow{b}|=| \overrightarrow{a} - \overrightarrow{b}|, \text{then} \theta = 90^{\circ}$

$\to$ when $\overrightarrow{a}+\overrightarrow{b}= \overrightarrow{a} - \overrightarrow{b}, \text{ then }, \overrightarrow{b} \text{ is null vector}$

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be any two vectors then

$\to$ $\overrightarrow{a}.\overrightarrow{b} = ab \text{ then } \theta = 0^{\circ}$

$\to$ $\overrightarrow{a}.\overrightarrow{b} = 0 \text{ then } \theta = 90^{\circ}$

$\to$ $\overrightarrow{a}.\overrightarrow{b} = -ab \text{ then } \theta = 180^{\circ}$

$\to$ $\overrightarrow{a} \times \overrightarrow{b} = 0 \text{ then } \theta = 0^{\circ} or 180^{\circ}$

$\to$ $\overrightarrow{a}.\overrightarrow{b} = |\overrightarrow{a} \times \overrightarrow{b}| \text{ then } \theta = 45^{\circ}$

$\to$ The angle between $\overrightarrow{a} \times \overrightarrow{b}$ and $\overrightarrow{b} \times \overrightarrow{a}$ is $180^{\circ}$

Examples of the scalar product

$\to$ Work done = $\overrightarrow{F}.\overrightarrow{s}$

$\to$ Power = $\overrightarrow{F}.\overrightarrow{v}$

$\to$ Electric flux = $\overrightarrow{E}.\overrightarrow{A}$

$\to$ Magnetic flux = $\overrightarrow{B}.\overrightarrow{A}$

Examples of the vector product

$\to$ Torque = $\overrightarrow{r} \times \overrightarrow{F}$

$\to$ velocity = $\overrightarrow{\omega} \times \overrightarrow{r}$

$\to$ Angular momentum = $\overrightarrow{r} \times \overrightarrow{p}$

$\to$ centripetal acceleration = $\overrightarrow{\omega} \times \overrightarrow{v}$

$\to$ tangential acceleration = $\overrightarrow{\alpha} \times \overrightarrow{r}$

Let $\hat{i}, \hat{j}$ and $\hat{k}$ be unit vectors along x-axis, y-axis and z-axis respectively. Then,

$\to \hat{i}.\hat{i} = \hat{j}.\hat{j} = \hat{k}.\hat{k} = 1$

$\to \hat{i}.\hat{j} = \hat{j}.\hat{k} = \hat{k}.\hat{j} = 0$

$\to \hat{j}.\hat{i} = \hat{k}.\hat{j} = \hat{j}.\hat{k} = 0$

$\to \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$

$\to \hat{i} \times \hat{j} = - \hat{j} \times \hat{i} = \hat{k}$

$\to \hat{j} \times \hat{k} = - \hat{k} \times \hat{j} = \hat{i}$

$\to \hat{k} \times \hat{i} = - \hat{i} \times \hat{k} = \hat{j}$

Minimum number of

$\to$ non zero collinear vectors giving zero resultant is 2.

$\to$ noncollinear coplanar vectors giving zero resultant is 3.

$\to$ non-coplanar vectors giving zero resultant is 4.

If $\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} =0$ then $\overrightarrow{A} \times \overrightarrow{B} = \overrightarrow{B} \times \overrightarrow{C} = \overrightarrow{C} \times \overrightarrow{A}$

Area of Parallelogram and Triangle

$\to$ The area of $\textbf{triangle}$ having $\overrightarrow{A}$ and $\overrightarrow{B}$ as $\textbf{adjecent sides}$ is $\dfrac{1}{2} |\overrightarrow{A} \times \overrightarrow{B} |$

$\to$ The area of $\textbf{Parallelogram}$ having $\overrightarrow{A}$ and $\overrightarrow{B}$ as $\textbf{adjecent sides}$ is $|\overrightarrow{A} \times \overrightarrow{B} |$

$\to$ The area of $\textbf{Parallelogram}$ having $\overrightarrow{A}$ and $\overrightarrow{B}$ as $\textbf{diagonals}$ is $\dfrac{1}{2}|\overrightarrow{A} \times \overrightarrow{B} |$

For three vectors of a different magnitude to be able to give zero resultant, they should form a triangle. i.e.

$\to$ The largest magnitude should be greater or equal to the sum of the other two magnitudes.

$\to$ eg, The vectors with magnitudes $10,20,40$ can not give zero resultant.

$\to$ and, The vectors with magnitudes $6,8,10$ may give zero resultant.

A vector of magnitude A is rotated by angle $\theta$ then, the change is vector is $$2A \sin \dfrac{\theta}{2}