Revision: Class 11 >> Concept of Vector and Motion in a Plane NEET (UG) Concept of Vector and Motion in a Plane

The motion of an object in which its position changes along two mutually perpendicular axes (X and Y) simultaneously, such that it requires two coordinates to describe its position at any given instant, is called Motion in a Plane (Two-Dimensional Motion).

When a vector is resolved into components along mutually perpendicular directions (like x and y axes in 2D, or x, y, and z axes in 3D), these components are called rectangular or Cartesian components.

The splitting vectors obtained when a single vector is resolved into two or more vectors in different directions are called component vectors.

The values of cos⁡α, cos⁡β, and cosγ which are the cosines of the angles subtended by the rectangular components with the given vector are called direction cosines of a vector.

A vector \[\vec V\] can be expressed as the sum of two or more vectors along fixed directions. This process is known as vector resolution.

OR

The process of splitting a single vector into two or more vectors in different directions which together produce same effect as produced by the single vector alone is called resolution of vector.

The total maximum horizontal distance travelled by a projectile from the point of projection to the point where it hits the ground is called the horizontal range (R).

An object in flight after being thrown with some velocity that follows a curved path under the action of gravity is called a projectile.

The maximum vertical height reached by the projectile — i.e., the distance travelled along the vertical (y) direction up to the highest point — is called the maximum height (H).

The total time for which the projectile remains in the air — from the moment it is projected to the moment it returns to the same level — is called the time of flight (T).

The time taken by the projectile to travel from the point of projection to the maximum height is called the time of ascent (t_A).

The time taken by the projectile to travel from the maximum height back to the ground is called the time of descent (t_D).

The angle traced out by the radius vector at the centre of the circular path in a given time, expressed as Δθ = θ₂ − θ₁​, is called angular displacement.

Angular velocity of a particle is the rate of change of angular displacement.

When a particle moves with a constant speed in a circular path, its motion is said to be uniform circular motion.

OR

The motion of a body moving with constant speed along a circular path is called uniform circular motion.

When a particle moves with a constant speed in a circular path, its motion is said to be the uniform circular motion.

The rate of change of angular displacement of a body undergoing circular motion is called angular velocity.

The rate of change of angular velocity of a body is called angular acceleration.

The component of acceleration directed towards the centre of the circular path is called centripetal acceleration (or radial acceleration).

The time taken by a particle performing uniform circular motion to complete one revolution is called time period.

The force directed towards the centre along the radius, required to keep a body moving along a circular path at constant speed, is called centripetal force.

If α, β, and γ are the angles subtended by the rectangular components with the given vector, then:

cos α = \[\frac {A_x}{A}\], cos β = \[\frac {A_y}{A}\], cos γ = \[\frac {A_z}{A}\]

The sum of squares of all direction cosines is always equal to 1:

cos²α + cos²β + cos²γ = 1

When a vector \[\vec A\] is resolved into three-dimensional rectangular components, it is given by:

The magnitude of vector \[\vec A\] resolved into three-dimensional components is:

A = \[\sqrt{A_x^2+A_y^2+A_z^2}\]​​

1. Component Method: Resultant R = A + B is found as R_x = A_x + B_x​, R_y = A_y + B_y, R_z = A_z + B_z​, giving R = R_x\[\hat i\] + R_y\[\hat j\] + R_z\[\hat k\].

2. Laws of Addition: Triangle law (head-to-tail), Parallelogram law (tail-to-tail, diagonal = resultant), and Polygon law (for multiple vectors, closing side = resultant).

3. Magnitude (Addition): When A and B are at angle θ, R = \[\sqrt{A^2+B^2+2AB\cos\theta}\].

4. Magnitude (Subtraction): Change the sign to minus — ∣R∣ = ​\[\sqrt{A^2+B^2-2AB\cos\theta}\].

5. Direction of Resultant: tan⁡α = \[\frac{B\sin\theta}{A+B\cos\theta}\] for addition; tan⁡β = \[\frac{B\sin\theta}{A-B\cos\theta}\] for subtraction.

• In UCM, speed is constant, but velocity continuously changes direction, always remaining tangential to the path.
• Angular displacement is the angle swept by the radius vector; angular velocity is its rate of change.
• Even at constant speed, centripetal acceleration is never zero — it always acts towards the centre of the circular path.
• Centripetal force is always directed towards the centre and is essential to maintain circular motion — it does no work on the body.
• If speed is constant in circular motion, tangential acceleration = 0, but radial acceleration ≠ 0.