No, only vectors representing physical quantities, of same nature can be added together.

No, pressure is always taken to be normal to the plane of the area on which it is acting. As this direction is unique, it does need any specification. So pressure is not a vector.

Let the resultant make angle \(\beta\) with the force P.
In first case, \(\tan\beta=\frac{2P\sin\theta}{P+2P\cos\theta}\)
In second case, \(\tan\beta=\frac{(2P+20)\sin\theta}{2P+(2P+20)\cos\theta}\)
Hence \(\frac{(2P+20)\sin\theta}{2P+(2P+20)\cos\theta}=\frac{2P\sin\theta}{P+2P\cos\theta}\)
 or \(\frac{2P\sin\theta}{P+2P\cos\theta}=\frac{20\sin\theta}{P+20\cos\theta}\)
From the above equation, 2P = 20   (or) P = 10 N.

\(\left| \vec { A } +\vec { B } \right| =\left| \vec { A } -\vec { B } \right| \)
Squaring both sides  \(A^2+B^2+2\vec{A}.\vec{B}=A^2-2\vec{A}.\vec{B}+B^2\)
\(4\vec{A}.\vec{B}=0\) (or) \(\vec{A}.\vec{B}=0 i.e, \vec{A}\bot\vec{B}\)
Thus, when the two vectors are equal in magnitude and perpendicular to each other, then he sum and difference of two vectors will be equal in magnitude.

Given: \(\vec{A}=\hat{i}+2\hat{j}+2\hat{k}\)   \(\vec{B}=\hat{i}+3\hat{j}+6\hat{k}\)
Dot product:  A = \(\sqrt{(1)^2+(2)^2+(2)^2}\) = 3
B = \(\sqrt{(1)^2+(3)^2+(6)^2}\) = \(\sqrt{46}\)
\(\vec{A}.\vec{B}=(\hat{i}+2\hat{j}+2\hat{k}).(\hat{i}+3\hat{j}+6\hat{k})\)
= 1\(\times\)1 + 2\(\times\)3 + 2\(\times\) 6 = 19.

Let e be the angle between \(\vec{A}\) and \(\vec{B}\). Thus the resultant of \(\vec{A}\) and \(\vec{B}\) is given by
 \(\vec{R}=\sqrt{A^2+B^2+2AB\cos\theta}\)
or R² = A² + B² + 2AB cos\(\theta\)
Now greatest value of cos\(\theta\) is + 1.
Greatest value of R² is (A² + B² + 2AB) or (A + B)² i.e., the greatest value of R is (A + B).
The least value of cos\(\theta\) is -1.
So the least value of R² is (A2 + B² - 2AB) or (A - B)² i.e., the least value of R is (A - B).

If the velocity changes in both speed and direction during the circular motion, we get non uniform circular motion.

When a point object covers equal distances on the circumference of the circle in equal intervals of time.

Degree is the unit of measurement which is used to determine the size of an angle.

Given: Angle between the forces, \(\theta =60^o\); Resultant R = 20\(\sqrt{3}\)N; P = Q = P (say) = ?
Resultant R = \(\sqrt{P^2+Q^2+2PQ\cos\theta}\)
= \(\sqrt{P^2+P^2+2P.p\cos\theta}\)
= \(\sqrt{2P^2+2P^2+\frac{1}{2}}=P\sqrt{3}\)
But the resultant force is = 20\(\sqrt{3}\)
20 \(\sqrt{3}\) = p \(\sqrt{3}\)
P = 20 N.

Let the force be, \(\vec{F}\) . Let angle of inclination be \(\theta\).
\(\therefore\) Horizontal component F_H = F cos \(\theta\)
Vertical component F_v = F sin \(\theta\)
Given: F_H = 40 kg. wt.
Angle \(\theta\) = 60°
F_H = F cos 60° = 40 kg wt
F = \(\frac{40}{60^o}=\frac{40}{\frac{1}{2}}\) = 80 kg wt
Vertical component F_v = F sin \(\theta\)
= 80 sin 60^o = 80\(\times\)\(\frac{\sqrt{3}}{2}=40\sqrt{3}\)
= 40\(\times\)1.732 = 69.28 kg wt.
Vertical component of the force = 69.28 kg wt.

Given   \(\vec{a}=3\hat{i}+2\hat{j}-\hat{k}\),  \(\vec{b}=\hat{i}+\hat{j}-3\hat{k}\)         
\(\vec{a}-\vec{b}=(3\hat{i}+2\hat{j}-\hat{k})-(\hat{i}+\hat{j}+3\hat{k})\)
\(= 3\hat{i}+2\hat{j}-\hat{k}-\hat{i}-\hat{j}-3\hat{k}\)
\(= 2\hat{i}+\hat{j}-4\hat{k}\)

Given :     \(\vec{a}=3\hat{i}+2\hat{j}-\hat{k}\),  \(\vec{b}=\hat{i}+\hat{j}-3\hat{k}\)
\(\vec{a}+\vec{b}=(3\hat{i}+2\hat{j}-\hat{k})+(\hat{i}+\hat{j}+3\hat{k})\)
\(= 3\hat{i}+2\hat{j}-\hat{k}+\hat{i}+\hat{j}+3\hat{k}\)
\(= 4\hat{i}+3\hat{j}+2\hat{k}\)

No, because equations of motions are applicable as long as the acceleration is uniform.

The path followed by the particle is called its trajectory.

The motion of a body falling towards the Earth from a small altitude (h « R), purely under the force of gravity is called free fall.

In accelerated motion, if the change in velocity of an object per unit time is same (constant) then the object is said to be moving with uniformly accelerated motion.
On the other hand, if the change in velocity per unit time is different at different times, then the object is said to be moving with non-uniform accelerated motion.

When two objects A and B are moving with different velocities, then the velocity of one object A with respect to another object B is called relative velocity of object A with respect to B.

Consider a butterfly and a stone, both moving towards you with the same velocity 5 ms^-1. If both hit your body, the effects will not be the same. The effects not only depend upon the velocity but also on the mass. The stone has greater mass compared to the butterfly. The momentum of the stone is thus greater than the momentum of the butterfly.

Average velocity: The average velocity is defined as ratio of the displacement vector to the corresponding time interval.
\(\vec{v_{avg}}=\frac{\Delta\vec{r}}{\Delta t}\)
It is a vector quantity. The direction of average velocity is in the direction of the displacement vector (AY).
Average speed: The average speed is defined as the ratio of total path length travelled by the particle in a time interval.
Average speed = Total path length / total time
It is a scalar quantity. Doesn't possess direction.

Any physical quantity is represented by a "function" in mathematics. Take the example of temperature T. We know that the temperature of the surroundings is changing throughout the day. It increases till noon and decreases in the evening. At any time "t" the temperature T has a unique value. Mathematically this variation can be represented by the notation 'T (t)' and it should be called "temperature as a function of time". It implies that if the value of 't' is given, then the function "T (t)" will give the value of the temperature at that time 't'.
Eg: Consider a function f(x) = x² Sometimes it is also represented as y = x². Here y is called the dependent .variable and x is called independent variable. It means as x changes, y also changes. Once a physical quantity is represented by a function, one can study the variation of the function over time or over the independent variable on which the quantity depends.

A vector which denotes the position of a particle at any instant of time, with respect to origin of coordinate system.
The position vector \(\vec{r}\) of the particle at a point P is given by \(\vec{r}=x\vec{i}+y\vec{j}+z\vec{k}\)

Similarly the subtraction of two vectors is equivalent to subtracting the corresponding x, y and z components:
\(\vec{A}-\vec{B}=(A_x-B_{x})\hat{i}+(A_y-B_y)\hat{j}+(A_z-B_z)\hat{k}\)

Two vectors have same direction.

No.

If the x, y and z-axes are drawn in anticlockwise direction then the coordinate system is called as "right handed Cartesian coordinate system".

The branch of mechanics which deals with the motion of objects without taking force into account. The Greek word "kinema" means "motion".

(i) Horizontal component of velocity.
(ii) At an angle of 45° to the horizontal.

(i) In uniform circular motion, speed remains constant but velocity changes continuously due to change in its direction.
(ii) Even though the velocity is tangential at every point in the circle, the acceleration is acting towards the center of the circle. This is called centripetal acceleration. It always points towards the center of the circle.

The rate of change of angular velocity is called angular acceleration.
\(\overrightarrow { a } =\frac { d\overrightarrow { \omega } }{ dt } \)

(i) For a given initial speed u, the maximum possible range is reached when sin 28 is maximum, sin 2 \(\theta\) = 1. This implies 2\(\theta\) = \(\pi\)/2.
R = \(\frac { { u }^{ 2 }sin\ 2\theta }{ g } \ or\ \theta =\frac { \pi }{ 4 } \)
(ii) This means that if the particle is projected at 45 degrees with respect to horizontal, it attains maximum range, given by
\({ R }_{ max }=\frac { { u }^{ 2 } }{ g } \)

Time of flight (T_f) The total time taken by the projectile from the point of projection till it hits the horizontal plane is called time of flight.

(i) When the projectile hits the ground after initially thrown horizontally from the top of tower of height h, the time of flight is
\(t=\sqrt { \frac { 2h }{ g } } \)
(ii) The horizontal component velocity of the projectile remains the same i.e V_x = u.
(iii) The vertical component velocity of the projectile at time T is
\({ v }_{ y }=gt=g\sqrt { \frac { 2h }{ g } } =\sqrt { 2gh } \quad \)
(iv) The speed of the particle when it reaches the ground is
\(v=\sqrt { { u }^{ 2 }+2gh } \) 

(i) A uniform velocity in the horizontal direction, which will not change provided there is no air resistance.
(ii) A uniformly changing velocity (i.e., increasing or decreasing) in the vertical direction.

(i) The velocity can be found from the area under the acceleration-time graph.
\(v=\int _{ { t }_{ 1 } }^{ { t }_{ 2 } }{ adt } \quad \)
(ii) From \(\frac { dv }{ dt } =a,\) we have dv = adt; hence for an initial time t₁ and final time t₂.

(i) If an object changes its velocity from \(\overrightarrow { { v }_{ 1 } } to\ \overrightarrow { { v }_{ 2 } } \) in a time interval \(\Delta t={ t }_{ 2 }-{ t }_{ 1 },\) then the average acceleration is defined as the ratio of change in velocity over the time interval \(\Delta t={ t }_{ 2 }-{ t }_{ 1 }\)
\(\overrightarrow { { a }_{ avg } } =\frac { \overrightarrow { { v }_{ 2 } } -\overrightarrow { { v }_{ 1 } } }{ \overrightarrow { { t }_{ 2 } } -\overrightarrow { { t }_{ 1 } } } =\frac { \Delta \overrightarrow { v } }{ \Delta t } \)
(ii) Average acceleration is a vector quantity in the same direction as the vector \(\Delta v\).

If the velocity is decreasing with respect to time then the acceleration is called retardation or deceleration (or) negative acceleration.

When two objects A and B are moving with different velocities, then the velocity of one object A with respect to another object B is called relative velocity of object A with respect to B.

(i) The linear momentum or simply momentum of a particle is defined as product of mass with velocity. It is denoted as '\(\overrightarrow { p } \)'. Momentum is also a vector quantity.
\(\overrightarrow { p } =m\overrightarrow { v } \)
(ii) The direction of momentum is also in the direction of velocity, and the magnitude of momentum is equal to product of mass and speed of the particle.

Calculus is the branch of mathematics used to analyse the change of any quantity.

(i) Any physical quantity is represented by a "function" in mathematics. For example:- temperature T. We know that the temperature varies with time.
(ii) Mathematically this variation can be represented by the notation 'T (t)' and it should be called "temperature as a function of time".
(iii) It implies that if the value of 't' is given, then the function "T (t)" will give the value of the temperature at that time 't'.

(i) The Distance travelled by an object in motion in a given time is never negative or zero, it is always positive.
(ii) The displacement of an object, in a given time can be positive, zero or negative.
(iii) The displacement of an object can be equal or less than the distance travelled but never greater than distance travelled.
(iv) The distance covered by an object between two positions can have many values, but the displacement between them has only one value (in magnitude).

(i) It is a vector which denotes the position of a particle at any instant of time, with respect to some reference frame or coordinate system.

(ii) The position vector \(\overrightarrow { r } \)of the particle at a point P is given by \(\overrightarrow { r } =x\overrightarrow { i } +y\overrightarrow { j } +z\overrightarrow { k } \) where x, y and z are components of \(\overrightarrow { r } \).

(i) Torque \(\overrightarrow { t } =\overrightarrow { r } \times \overrightarrow { F } \). where F is Force and \(\overrightarrow { r } \) is position vector of a particle
(ii) Angular momentum \(\overrightarrow { L } =\overrightarrow { r } \times \overrightarrow { p }\ where\ \overrightarrow { p } \)  is the linear momentum 
(iii) Linear Velocity \(\overrightarrow { v } =\overrightarrow { \omega } \times \overrightarrow { r } \)  where \(\overrightarrow { \omega } \) is angular velocity.

It is the process of splitting a vector into two or more vectors in such a way that their combined effect is same as that of the given vector.

i) The vector product of two vectors is not commutative, i.e., \(\overrightarrow { A } \times \overrightarrow { B } \neq \overrightarrow { B } \times \overrightarrow { A } \) But., \(\overrightarrow { A } \times \overrightarrow { B } =-[\overrightarrow { B } \times \overrightarrow { A } ]\)
(ii) Here it is worthwhile to note that \(|\overrightarrow { A } \times \overrightarrow { B } |=[\overrightarrow { B } \times \overrightarrow { A } ]\)= AB sin \(\theta\)  i.e., in the case of the product vectors \(\overrightarrow { A } \) \(\times\)\(\overrightarrow { B } \) and \(\overrightarrow { B } \) \(\times\) \(\overrightarrow { A } \) , the magnitudes are equal but directions are opposite to each other.

\(|3\hat { i } -2\hat { j } +\sqrt { 3 } \hat { k } |=\sqrt { { 3 }^{ 2 }+{ 2 }^{ 2 }+{ \sqrt { 3 } }^{ 2 } } \)
\(\sqrt { 9+4+3 } =\sqrt { 16 } =4\)

The length of a vector is called magnitude of the vector. It is always a positive quantity. The magnitude or norm is denoted by \(\left| \overrightarrow { A } \right| \) .

A coordinate system and the position of an object is described relative to it, then such a coordinate system is called frame of reference.

Kinematics is the branch of mechanics which deals with the motion of objects without taking force into account. The Greek word "kinema" means "motion".

(i) Air resistance is neglected.
(ii) The effect due to rotation of Earth and curvature of Earth is negligible.
(iii) The acceleration due to gravity is constant in magnitude and direction at all points of the motion of the projectile.