Lecture on Vectors

Introduction to Vectors

• Objective: Identify which quantities are vectors vs. scalars.
• Quantities discussed:
  • Displacement
  • Velocity
  • Acceleration
  • Mass
  • Force

Scalar vs. Vector Quantities

• Scalar Quantities: Have magnitude only, no direction.
  • Example: Temperature (e.g., 80°F)
• Vector Quantities: Have both magnitude and direction.
  • Example: Force (e.g., 100 N at 30° above x-axis)

Examples of Scalars and Vectors

• Scalars:
  • Distance
  • Speed
  • Mass
• Vectors:
  • Displacement
  • Velocity
  • Acceleration
  • Force

Key Concepts

• Displacement: Distance with direction.
• Velocity: Speed with direction.
• Acceleration: Change in velocity.

Problem: Identifying Non-Vector Quantity

• Mass is a scalar quantity (e.g., 10 kg).
• Correct answer: Mass, as it has magnitude only, no direction.

Calculating Components of a Force Vector

• Given: A force vector with magnitude 100 N at 30° above x-axis.
• Calculating x and y components using trigonometry.

Trigonometric Principles (SOHCAHTOA)

• Sine (SO):
  • Formula: $\sin(\theta) = \frac{opposite}{hypotenuse}$
  • $F_y = F \cdot \sin(\theta)$
• Cosine (CA):
  • Formula: $\cos(\theta) = \frac{adjacent}{hypotenuse}$
  • $F_x = F \cdot \cos(\theta)$
• Tangent (TO):
  • Formula: $\tan(\theta) = \frac{opposite}{adjacent}$
  • Angle: $\theta = \arctan(\frac{F_y}{F_x})$

Pythagorean Theorem

• For vectors: $c^2 = a^2 + b^2$
• Calculate magnitude of vector: $\sqrt{F_x^2 + F_y^2}$

Solving the Example Problem

• Given:
  • $F = 100 N$
  • $\theta = 30°$
• Components Calculation:
  • $F_x = 100 \cdot \cos(30°) = 86.6 N$
  • $F_y = 100 \cdot \sin(30°) = 50 N$

Expressing Vectors Using Unit Vectors

• Unit Vectors: i, j, k
  • i: x-axis
  • j: y-axis
  • k: z-axis
• Example Expression:
  • $F = 86.6i + 50j$

Conclusion

• Magnitude and direction are fundamental in vector calculations.
• Understanding of unit vectors aids in expressing vector components.