Vector

Vectors Overview

• Vectors are quantities that have both magnitude and direction.

• Common vector examples include displacement, velocity, and force.

1. Representation of Vectors

1.1 Graphical Representation

• Vectors can be depicted as arrows.

• The length of the arrow represents the magnitude, and the arrowhead shows the direction.

1.2 Algebraic Representation

• In 2D, a vector A can be represented in terms of its components:
  $extbf{A} = A<em>x extbf{i} + A</em>y extbf{j}$

• In 3D, it extends to:
  $extbf{A} = A<em>x extbf{i} + A</em>y extbf{j} + A_z extbf{k}$

2. Operations on Vectors

2.1 Addition

• Vectors are added tip-to-tail or by using components:
  $extbf{A} + extbf{B} = (A<em>x + B</em>x) extbf{i} + (A<em>y + B</em>y) extbf{j}$

2.2 Subtraction

• Similar to addition:
  $extbf{A} - extbf{B} = (A<em>x - B</em>x) extbf{i} + (A<em>y - B</em>y) extbf{j}$

2.3 Scalar Multiplication

• For a scalar $k$ and vector A:
  $k extbf{A} = kA<em>x extbf{i} + kA</em>y extbf{j}$

3. Dot Product and Cross Product

3.1 Dot Product (Scalar Product)

• Given two vectors A and B:
  $extbf{A} \bullet extbf{B} = | extbf{A}| | extbf{B}| ext{cos} heta$

• Provides a scalar quantity.

3.2 Cross Product (Vector Product)

• Given two vectors A and B:
  $extbf{A} imes extbf{B} = | extbf{A}| | extbf{B}| ext{sin} heta ext{n}$

• Produces a vector that is perpendicular to both A and B.

4. Applications of Vectors

• Used extensively in mechanics for representing forces, velocities, and other vector quantities.

• Essential for solving problems related to projectile motion, circular motion, and forces in equilibrium.

5. Properties of Vectors

• Vectors can be categorized as:

  • Equal vectors: Same magnitude and direction.

  • Unit vectors: Vectors of magnitude 1.

  • Null vector: A vector with zero magnitude, direction is undefined.

6. Special Vectors

• Position Vector: Indicates the position of a point in space relative to a reference.

• Velocity Vector: Represents the rate of change of position with respect to time.

• Acceleration Vector: Represents the rate of change of velocity with respect to time.

7. Theorems and Laws Involving Vectors

• Parallelogram Law of Vector Addition: If two vectors are represented as adjacent sides of a parallelogram, their resultant is represented by the diagonal.

• Triangle Law of Vector Addition: If two vectors are represented as two sides of a triangle taken in sequence, the resultant is represented by the closing side of the triangle.