class 11 physics chapter 2 vectors notes|| class 11 physics unit 2 vector handwritten notes|| class 11 vectors notes

• 
  

• 
  
  
  
  

• 
  

• For Handwritten notes:- Click Here To Download Notes

• 
  

• 
  

• 📘 Chapter: Vectors (Class 11 Physics)

  ---

  🔹 1. Physical Quantities: Scalars and Vectors

  ➤ Scalar Quantity

  A scalar quantity is one that has magnitude only and no direction.
  It can be completely described by a single numerical value and unit.

  Examples:

  • Mass (5 kg),

  • Temperature (30°C),

  • Time (10 s),

  • Distance,

  • Speed,

  • Energy,

  • Work

  ---

  ➤ Vector Quantity

  A vector quantity has both magnitude and direction.
  It is represented by an arrow:

  • Length = magnitude

  • Direction = direction of quantity

  Examples:

  • Displacement,

  • Velocity,

  • Acceleration,

  • Force,

  • Momentum

  ---

  🔹 2. Representation of a Vector

  ➤ Graphical Representation

  • Represented as an arrowed line:
    Start point = tail, endpoint = head

  • The length of the arrow ∝ magnitude

  • Angle with reference axis shows direction

  ➤ Symbol

  A vector A is written as →A or 𝐀
  Magnitude of A: $|\vec{A}|$ or simply A

  ---

  🔹 3. Types of Vectors

  Type	Definition
  Zero Vector (Null Vector)	A vector with zero magnitude, direction undefined. Noted as $\vec{0}$.
  Unit Vector	Vector with magnitude = 1, used to represent direction only. $$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$$
  Equal Vectors	Same magnitude and same direction
  Negative of a Vector	Same magnitude, opposite direction. If A, then −A is opposite.
  Collinear Vectors	Vectors lying along the same line or parallel
  Co-initial Vectors	Vectors with the same starting point
  Coplanar Vectors	Vectors lying in the same plane
  Position Vector	Vector from origin O to a point P: $\vec{r} = \vec{OP}$

  ---

  🔹 4. Addition of Vectors

  ➤ Triangle Law of Vector Addition

  If two vectors are placed head to tail, the third side of triangle (from start to end) is the resultant vector.

  ➤ Parallelogram Law of Vector Addition

  If vectors A and B originate from the same point and form a parallelogram, the diagonal gives the resultant.

  ➤ Formula for Resultant Vector

  If angle between A and B is θ:

  $$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$$

  ➤ Direction of Resultant

  $$\tan\phi = \frac{B\sin\theta}{A + B\cos\theta}$$

  Special Cases:

  • θ = 0° → R = A + B (same direction)

  • θ = 180° → R = |A − B| (opposite direction)

  • θ = 90° → $R = \sqrt{A^2 + B^2}$

  ---

  🔹 5. Subtraction of Vectors

  Subtracting vector B from A:

  $$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$$

  Reverse the direction of B and apply vector addition.

  ---

  🔹 6. Resolution of a Vector

  Breaking a vector into horizontal and vertical components (along x and y axes).
  Let a vector A make angle θ with x-axis:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

  So,

  $$\vec{A} = A_x \hat{i} + A_y \hat{j}$$

  Magnitude:

  $$|\vec{A}| = \sqrt{A_x^2 + A_y^2}$$

  Direction:

  $$\tan\theta = \frac{A_y}{A_x}$$

  ---

  🔹 7. Unit Vectors

  Unit vectors are used to indicate direction only.

  Unit Vector	Direction
  $\hat{i}$	Along x-axis
  $\hat{j}$	Along y-axis
  $\hat{k}$	Along z-axis

  Any vector in 3D:

  $$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$$

  ---

  🔹 8. Vector Multiplication

  ---

  (A) Dot Product (Scalar Product)

  $$\vec{A} \cdot \vec{B} = AB \cos\theta$$

  • Result is a scalar.

  • θ = angle between A and B

  Important Properties

  • $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$ (commutative)

  • $\vec{A} \cdot \vec{A} = A^2$

  • $\vec{A} \cdot \vec{B} = 0$ if A ⊥ B

  ---

  (B) Cross Product (Vector Product)

  $$\vec{A} \times \vec{B} = AB \sin\theta \ \hat{n}$$

  • Result is a vector

  • Direction: Perpendicular to the plane of A and B

  • Use Right-Hand Rule to find direction

  Important Properties

  • Not commutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$

  • Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$

  ---

  🔹 9. Important Vector Identities

  • $\vec{A} \cdot \vec{B} = 0$ if vectors are perpendicular

  • $\vec{A} \times \vec{B} = 0$ if vectors are parallel

  • $\vec{A} \cdot \vec{A} = |\vec{A}|^2$

  • $\vec{A} \times \vec{A} = 0$

  Unit Vector Cross Products:

  • $\hat{i} \times \hat{j} = \hat{k}$

  • $\hat{j} \times \hat{k} = \hat{i}$

  • $\hat{k} \times \hat{i} = \hat{j}$

  Unit Vector Dot Products:

  • $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$

  • $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

  ---

  🔹 10. Applications of Vectors in Physics

  • Describing motion (displacement, velocity, acceleration)

  • Newton’s laws (force is a vector)

  • Electric and magnetic fields

  • Momentum, torque, angular velocity

  • Work (dot product), torque (cross product)

  ---

  ✅ Summary of Important Formulas

  Concept	Formula
  Resultant of Two Vectors	$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$
  Direction of Resultant	$\tan\phi = \frac{B\sin\theta}{A + B\cos\theta}$
  Components of a Vector	$A_x = A \cos\theta, \ A_y = A \sin\theta$
  Dot Product	$\vec{A} \cdot \vec{B} = AB \cos\theta$
  Cross Product	$\vec{A} \times \vec{B} = AB \sin\theta \ \hat{n}$
  Magnitude from Components	$A = \sqrt{A_x^2 + A_y^2}$

  ---

  Let me know if you’d like:

  • A printable PDF version

  • A mind map or summary sheet

  • Some practice problems with solutions

  • Diagrams to understand each method

  Happy studying! 😊

download notes