SCALARS AND VECTORS

Scalars:

2. Examples:
   • Temperature
   • Time
   • Mass
   • Energy
   • Speed

Vectors:

1. Quantities with both magnitude and direction.
2. Represented graphically with arrows.
3. Examples:
   • Displacement
   • Velocity
   • Acceleration
   • Force
   • Momentum

Key Differences:

1. Direction: Vectors have direction, scalars don’t.
2. Addition: Vectors require vector addition, scalars use regular addition.
3. Multiplication: Vectors require dot product or cross product, scalars use regular multiplication.

Vector Operations:

1. Addition: Head-to-tail method or parallelogram law.
2. Subtraction: Reverse direction and add.
3. Scalar multiplication: Multiply magnitude, keep direction.
4. Dot product (scalar): a · b = |a| |b| cos(θ).
5. Cross product (vector): a × b = |a| |b| sin(θ) n.

Vector Properties:

1. Magnitude (length): |a|.
2. Direction: θ (angle with x-axis).
3. Unit vector: a / |a|.
4. Zero vector: 0 (no magnitude or direction).

Real-World Applications:

1. Physics: Motion, forces, energy.
2. Engineering: Stress, strain, fluid dynamics.
3. Navigation: Direction, velocity, acceleration.
4. Computer Graphics: 3D modeling, animation.

Mathematical Representations:

1. Component form: a = (x, y, z).
2. Matrix representation.
3. Polar coordinates: (r, θ).

Important Theorems:

1. Pythagorean theorem (vector addition).
2. Triangle inequality (vector magnitude).
3. Cauchy-Schwarz inequality (dot product).