Understanding Vectors

Definitions

• Scalar Quantity: Has magnitude but no direction. Example: Temperature.
  • e.g., 80 degrees Fahrenheit is a scalar because it doesn't involve direction.
• Vector Quantity: Has both magnitude and direction. Example: Force.
  • e.g., A force of 100 Newtons at 30 degrees above the x-axis is a vector because it specifies direction.

Scalar vs. Vector

• Distance is scalar, but Displacement is a vector (distance with direction).
• Speed is scalar, whereas Velocity is a vector (speed with direction).
• Mass is a scalar quantity; it only has magnitude, not direction. (Correct answer to the question posed at the beginning.)
• Acceleration and Force are vector quantities.

Vector Components

• A vector can be broken into x and y components.
• SOHCAHTOA is a mnemonic to remember trigonometric ratios:
  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent

Calculating Components

1. Y Component (FY): Y component of a force vector is magnitude of F * sine of angle θ.
2. X Component (FX): X component is magnitude of F * cosine of angle θ.
3. To calculate angles in vector problems, we use the arc tangent of FY/FX.
4. Magnitude of a Vector: Square root of (FX^2 + FY^2) based on Pythagorean theorem.

Example Problem: Force Vector

• Given: 100 Newton force at 30 degrees above the x-axis.
• Calculate X and Y components:
  • FX = 100 * cos(30) = 50√3 ≈ 86.6 Newtons
  • FY = 100 * sin(30) = 50 Newtons

Expressing Vectors with Unit Vectors

• Unit Vector: A vector of magnitude 1.
  • Represented by i (x-axis), j (y-axis), and k (z-axis).
• Original force vector F can be expressed as 86.6i + 50j.

Key Takeaway: Understanding vector components and how to express them is crucial for solving physics problems involving vectors.