🔖 Topics

  • Vectors
  • Dimensional Analysis

🎯 Objectives

  • Perform mathematical operations with two and three dimensional vectors
  • Use dimensional analysis to gain insight into the validity of equations

📋 Sequence

  • Scalars vs Vectors
  • Vector Properties
    • Coordinate Axes
    • Syntax (array vs. \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) vs. \(\hat{x}\), \(\hat{y}\), \(\hat{z}\))
    • Magnitude (Pythagorean Theorem)
    • Direction (Trigonometry)
  • Vector Arithmetic
    • Addition
    • Subtraction
    • Scalar Multiplication
  • Other Vector Operations
    • Dot Product
    • Cross Product
  • Unit Vectors and Normalization
  • Dimensional Analysis

🖥️ Animations, Simulations, Activities

N/A

📝 Practice Problems

Vector in a 2D Coordinate Grid: Let \(\vec{A}\) be a vector starting at (3, 5) and ending at (-1, 9).

  • Sketch out a diagram of the setup.
  • Write out \(\vec{A}\) in array format.
  • Write out \(\vec{A}\) in \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) format.
  • What is the magnitude of \(\vec{A}\)?
  • What angle does \(\vec{A}\) make with the positive x-axis?

Vector in a 3D Coordinate Grid: Let \(\vec{A}\) be a vector starting at (-7, 15, 11) and ending at (-1, -3, -5).

  • Sketch out a diagram of the setup.
  • Write out \(\vec{A}\) in array format.
  • Write out \(\vec{A}\) in \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) format.
  • What is the magnitude of \(\vec{A}\)?

Scalar Triple Product: Although we (probably) won’t use it in this class, the scalar triple product gives the volume of a parallelepiped defined by three vectors. It is defined by: \(V = \vec{a} \cdot (\vec{b} \times \vec{c})\)

  • Is the calculated value of the scalar triple product a scalar or a vector? How do you know?
  • Calculate the scalar triple product for \(\vec{a} = 5\hat{i} + 3\hat{j} - 2\hat{k}\), \(\vec{b} = 2\hat{i} + 3\hat{k}\), and \(\vec{c} = -13\hat{i} - 3\hat{j} - 1\hat{k}\)
  • What happens to the scalar triple product if \(\vec{b}\) is parallel to \(\vec{c}\)?
  • What happens to the scalar triple product if \(\vec{b}\) is anti-parallel to \(\vec{c}\)?
  • What happens to the scalar triple product if \(\vec{a}\) is perpendicular to \(\vec{b} \times \vec{c}\)?

Dimensional Analysis: After experimenting in the lab, I have determined a potential equation that relates the amount of liquid flowing in a pipe to the flow and pipe area. Use dimensional analysis to determine if the following equation could be valid:

\[V = C J A^2\]
  • Volumetric Flow Rate (V) = Volume Per Unit Time
  • Flow Velocity (J) = Length Covered by Liquid Per Unit Time
  • Cross-Sectional Vector Area (A) = Cross-Sectional Area of the Pipe

✅ Partial Solutions

N/A

📘 Connected Resources

  • Giambattista, Alan, et al. College Physics With an Integrated Approach to Forces and Kinematics. 5th ed., McGraw-Hill Education, 2020.