MOTION IN PLANE

Motion in a plane is called as motion in two dimensions e.g., projectile motion, circular motion etc. For the analysis of such motion our reference will be made of an origin and two co-ordinate axes X and Y. 

SCALAR QUANTITIES 

Those quantities which only have magnitude are known as scalar quantities.

for eg: distance , speed etc.

 VECTOR QUANTITIES

Those quantities which have magnitude as well as direction is known as vector quantities. Vector quantity is represent by arrow over the quantity in  which direction is represented by head and magnitude is represent by tail.

for eg: displacement, velocity, acceleration, force etc.

• Magnitude of vector: It is a scalar quantity, its also called mod of a vector and its given as

 a = xi + yj + zk 

 |a| = ( x² + y² + z²)^1/2

•  Unit Vector

A unit vector is a vector of unit magnitude and points in a particular direction. It is used to specify the direction only. Unit vector is represented by putting a cap (^) over the quantity.

•  Equal Vectors

•  Zero Vector( Null vector)

•  Negative of a Vector

• Parallel Vectors

•  Coplanar Vectors

Vectors are said to be coplanar if they lie in the same plane or they are parallel to the same plane, otherwise they are said to be non-coplanar vectors.

• Coinitial vectors

Those vectors which have same initial point are known as coinitial vectors.

• Orthogonal vectors

Those vectors which are perpendicular to each other are called orthogonal vectors.

•  Triangle Law of Vector Addition

If two vectors are represented both in magnitude and direction by the two sides of a triangle taken in the same order, then the resultant of these vectors is represented both in magnitude and direction by the third side of the triangle taken in the opposite order.

 

• Parallelogram Law of Vector Addition

If two vectors, acting simultaneously at a point, can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then the resultant is represented completely both in magnitude and direction by the diagonal of the parallelogram passing through that point.

If the resultant vector R subtends an angle β with vector A, then

                                    tan β = B sin θ / A + B cos θ 

Special cases: 

Case I:When the vectors are parallel to each other, i.e θ=0°,

R² = √(A² + 2ABcos0 + B²)
R² = √(A² + 2AB + B²)
R² = √(A + B)²

∴R = A + B (Maximum value of R) 

Case II:
When the vectors are perpendicular to each other, i.e θ=90°,
R² = √(A² + 2ABcos90 + B²)
∴R² = √(A² + B²)

Case III:
When the angle between the vectors (θ) is 180°. Then,
R² = √(A² + 2ABcos180 + B²)
R² = √(A² - 2AB + B²)
R² = √(A - B)²
∴ R = A - B (Minimum Value of R)

                        

• Polygon Law of Vector Addition

If a number of vectors are represented both in magnitude and direction by the sides of a polygon taken in the same order, then the resultant vector is represented both in magnitude and direction by the closing side of the polygon taken in the opposite order.