Lecture Notes: Vector Formulas in Physics

Introduction to Vectors

• Vector: Has both magnitude and direction.
• Components: Vector v has components v_x (x-component) and v_y (y-component).
• Notation:
  • Arrow on top: Indicates it's a vector.
  • Absolute value: Refers to the magnitude of the vector, which is a scalar (magnitude only, no direction).

Calculating Magnitude

• 2D Vector: Magnitude of v is ( \sqrt{v_x^2 + v_y^2} ).
• 3D Vector: Magnitude of v is ( \sqrt{v_x^2 + v_y^2 + v_z^2} ).

Vector Components

• 2D Vector: ( v = v_x \hat{i} + v_y \hat{j} ).
• 3D Vector: ( v = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} ).
• Standard Unit Vectors:
  • ( \hat{i} = (1, 0, 0) )
  • ( \hat{j} = (0, 1, 0) )
  • ( \hat{k} = (0, 0, 1) )

Dot Product

• Definition: Multiplies two vectors to give a scalar.
• Formula: ( \mathbf{A} \cdot \mathbf{B} = |A||B|\cos(\theta) )
• Components in 2D: ( A_xB_x + A_yB_y )
• Example: Calculating work using dot product (Work = Force x Displacement x ( \cos(\theta) )).

Cross Product

• Definition: Multiplies two vectors to give another vector.
• Formula: ( \mathbf{A} \times \mathbf{B} = |A||B|\sin(\theta) \mathbf{n} )
• 3x3 Determinant: Used to evaluate the cross product.
• Example: Torque (( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} )).

Unit Vectors

• Definition: A vector with magnitude 1, giving direction.
• Formula: Unit vector of ( \vec{v} ) is ( \hat{v} = \frac{\vec{v}}{|v|} ).
• Position Vector: From origin to a point ( (x, y, z) ).

Applications

• Magnetic Force: ( \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) )
• Electric Field: ( \mathbf{E} = \frac{kq}{r^2} \hat{r} )

Additional Resources

• Links provided for more formulas and video tutorials for further learning on vectors.

These notes summarize the critical concepts and formulas related to vectors as discussed in the lecture, focusing on their application in physics.