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