Vectors - Class 11 Physics Lecture Notes

Physical Quantities Classification

• Vector Quantity:

  • Has magnitude and direction.
  • Denoted with an arrow over a letter.
  • Examples: Displacement, velocity, acceleration, force.
  • Note: Electric current and pressure have magnitude and direction but do not follow vector algebra (they are scalar).

• Scalar Quantity:

  • Has only magnitude, no direction.
  • Examples: Mass, length, time, distance, speed.

Representation of a Vector

• A vector (\overrightarrow{a}) can be expressed as: [ a\overrightarrow{a} = a_1 \widehat{i} + a_2 \widehat{j} + a_3 \widehat{k} ]
  • Components (a_1, a_2, a_3) along X, Y, and Z axes.
  • (\widehat{i}, \widehat{j}, \widehat{k}) are unit vectors.

Modulus/Magnitude of a Vector

• Given by: [ |\overrightarrow{a}| = \sqrt{{x}^2 + {y}^2 + {z}^2} ]

Types of Vectors

1. Unit Vector: Magnitude = 1. Denoted by a cap over a letter.
2. Null Vector: Magnitude = 0.
3. Parallel Vectors: Same/different magnitudes, same direction.
4. Anti-parallel Vectors: Same/different magnitudes, opposite directions.
5. Equal Vectors: Same magnitude and direction.
6. Negative Vector: Same magnitude, opposite direction.
7. Collinear Vectors: Act along the same line.
8. Coplanar Vectors: Act on the same plane.
9. Position Vectors: Origin is the initial point.
10. Co-initial Vectors: Have the same starting point.
11. Co-terminal Vectors: Have the same ending point.
12. Orthogonal Vectors: Perpendicular to each other.

Composition of Vectors

• Resultant Vector: A single vector obtained from two/more vectors.

Triangle Law of Vector Addition

• Statement: Two vectors represented by two sides of a triangle; the third side represents the resultant.

Parallelogram Law of Vector Addition

• Statement: Two vectors represented by adjacent sides of a parallelogram; the diagonal represents the resultant.

Special Cases

• Same direction (\theta = 0): (R = P + Q)
• Opposite direction (\theta = 180): (R = |P - Q|)
• Perpendicular (\theta = 90): (R = \sqrt{P^2 + Q^2})
• Equal magnitudes: Calculate using cosine values.

Polygon Law of Vector Addition

• Vectors represented by sides of a polygon; the closing side represents the resultant.

Subtraction of Vectors

• Using parallelogram law to find (\overrightarrow{A} - \overrightarrow{B}).

Resolution of a Vector

• Splitting a vector into perpendicular components.

Multiplication of Vectors

By a Number

• Result is (n\overrightarrow{a}).

By a Scalar

• Result is a new vector with the same direction.

By Another Vector

Dot Product (Scalar Product)

• (\overrightarrow{a} \cdot \overrightarrow{b} = ab\cos\theta)
• Scalar result.
• Maximum when (\theta = 0), zero when (\theta = 90).

Cross Product (Vector Product)

• (\overrightarrow{a} \times \overrightarrow{b} = ab\sin\theta\widehat{n})
• Vector result, perpendicular to both vectors.
• Maximum when (\theta = 90), zero when (\theta = 0).

Properties

• Dot product is commutative and distributive.
• Cross product is distributive but not commutative.

Numerical Problems Examples

1. Displacement Calculation
2. Surveying a Cave
3. Rocket Thrust Calculation

Link to Kinematics Notes