(a) v= u+at

(b) s= ut+1/2 at²

(c) v²=u²+2as

Here u is the initial velocity, v is the final velocity, a is the acceleration , s is the displacement travelled by the body and t is the time.

Note: Take ‘+ve’ sign for a when the body accelerates and takes ‘–ve’ sign when the body decelerates.

• The displacement by the body in n^th second is given by,

s_n= u + a/2 (2n-1)

• Position-time (x vs t), velocity-time (v vs t) and acceleration-time (a vs t) graph for motion in one-dimension: 

(i) Variation of displacement (x), velocity (v) and acceleration (a) with respect to time for different types of motion.  

• Scalar Quantities:- Scalar quantities are those quantities which require only magnitude for their complete specification.(e.g-mass, length, volume, density)

• Vector Quantities:- Vector quantities are those quantities which require magnitude as well as direction for their complete specification. (e.g-displacement, velocity, acceleration, force)

• Null Vector (Zero Vectors):- It is a vector having zero magnitude and an arbitrary direction.

When a null vector is added or subtracted from a given vector the resultant vector is same as the given vector.

Dot product of a null vector with any arbitrary is always zero. Cross product of a null vector with any other vector is also a null vector.

• Collinear vector:- Vectors having a common line of action are called collinear vector. There are two types.

Parallel vector (θ=0°):- Two vectors acting along same direction are called parallel vectors.

Anti parallel vector (θ=180°):-Two vectors which are directed in opposite directions are called anti-parallel vectors.

• Co-planar vectors- Vectors situated in one plane, irrespective of their directions, are known as co-planar vectors.

• Vector addition:-

Vector addition is commutative-  

Vector addition is associative-   

Vector addition is distributive-   

• Triangles Law of Vector addition:- If two vectors are represented by two sides of a triangle, taken in the same order, then their resultant in represented by the third side of the triangle taken in opposite order.

       Magnitude of resultant vector :-

       R=√(A²+B²+2ABcosθ)

        Here θ is the angle between  and .

        If β is the angle between  and ,

 then,

• If three vectors acting simultaneously on a particle can be represented by the three sides of a triangle taken in the same order, then the particle will remain in equilibrium.

So,  

• Parallelogram law of vector addition:-

R=√(A²+B²+2ABcosθ),

 

Cases 1:- When, θ=0°, then,

 R= A+B (maximum), β=0°

Cases 2:- When, θ=180°, then,

R= A-B (minimum), β=0°

Cases 3:- When, θ=90°, then,

R=√(A²+B²), β = tan^-1 (B/A)

• The process of subtracting one vector from another is equivalent to adding, vectorially, the negative of the vector to be subtracted. 

So,   

• Resolution of vector in a plane:-

• Product of two vectors:-

(a) Dot product or scalar product:-

 , 

Here A is the magnitude of , B is the magnitude of  and θ  is the angle between  and .

(i) Perpendicular vector:-

(ii) Collinear vector:-

When, Parallel vector (θ=0°),

When, Anti parallel vector (θ=180°),

(b) Cross product or Vector product:-