🔖 Topics

  • Right-handed coordinate systems
  • Review of trigonometry
  • Review of vectors
  • Conceptual notion of fields

🎯 Objectives

  • Be able to draw and utilize a right-handed coordinate system
  • Calculate the side lengths of a right triangle using the Pythagorean theorem
  • Calculate the angles around a right triangle using sine, cosine, and tangent
  • Perform basic operations on vectors (addition, subtraction, scalar multiplication, etc.)
  • Calculate dot and cross products of vectors
  • Describe a scalar and vector field, conceptually

📋 Sequence

  • Right-handed coordinate systems
  • Review of trigonometry
    • Right triangles
    • Pythagorean theorem
    • Sine, cosine, and tangent functions
    • Inverse sine, cosine, and tangent functions
  • Review of vectors
    • Scalar multiplication
    • Vector addition
    • Vector subtraction
    • Dot product
    • Cross product
    • Magnitude
    • Unit vectors
    • Vector components
  • Fields
    • Definition
    • Scalar fields
    • Vector fields

🖥️ Animations, Simulations, Activities

📝 Practice Problems

Vectors Practice

  1. Perform the following calculations. Then, sketch the initial and final vectors:

    \[2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} = \: ?\] \[-2 \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \: ?\] \[\begin{bmatrix} 4 \\ -3 \\ 5 \end{bmatrix} + \begin{bmatrix} 5 \\ 1 \\ -5 \end{bmatrix} = \: ?\] \[(3\hat{x} + 6\hat{y}) - (4\hat{x} + 2\hat{y}) = \: ?\] \[(2\hat{i} + 2\hat{j} + 2\hat{k}) - (7\hat{i} - 2\hat{j} + 3\hat{k}) = \: ?\]

  2. A vector, \(\vec{A}\) points in the \(+\hat{x}\) direction. Draw three choices for a vector, \(\vec{B}\), such that \(\vec{A} + \vec{B}\) points in the \(+\vec{y}\) direction.

  3. Two vectors are given by \(\vec{A} = 3\hat{x} + 7\hat{y}\) and \(\vec{B} = 4\hat{x} + 2\hat{y}\).

    • Calculate the dot product of these vectors.
    • Calculate the angle between the two original vectors.
    • Sketch out the two original vectors and their dot product.

  4. Two vectors are given by \(\vec{A} = -2\hat{x} + 6\hat{y} - 4\hat{z}\) and \(\vec{B} = 2\hat{x} + 2\hat{y} - 3\hat{z}\).

    • Calculate the cross product of these vectors.
    • Sketch out the two original vectors and their cross product

  5. Consider the vector \(\vec{A} = 5 \hat{x} - 2\hat{y}\).

    • What is its magnitude?
    • What is its direction (i.e. angle, measured from the positive x-axis)?
    • Write the formula for a unit vector pointing in the same direction as \(\vec{A}\).

  6. A sled at rest is suddenly pulled in three different directions at the same time, but it goes nowhere. Paul pulls to the northeast with a force of 50 lb. Johnny pulls at an angle of \(45^\circ\) south of due west with a force of 65 lb. Connie pulls with a force to be determined.

    • Sketch a diagram of the system.
    • Express the boys’ two forces in terms of \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\).
    • Determine the third force (from Connie).

Trigonometry Practice

  1. A vector has a magnitude of 30 Newtons and a direction of \(12^\circ\), counter-clockwise from the positive x-axis. Write this vector as a column vector.

  2. Convert \(25^\circ\) to radians.

  3. A zip wire runs between two posts which are 25 meters apart. The zip wire is at an angle of \(10^\circ\) with respect to the horizontal. Calculate the length of the zip wire.

  4. Express \(tan(\theta)\) in terms of \(sin(\theta)\) and \(cos(\theta)\).

Fields Practice

  1. Sketch the gravitational field around the planet Earth when you are zoomed out on a planetary scale.
  2. Sketch the approximate gravitational field near the surface of the Earth (this is the assumption we made for most of PHY 121).

✅ Partial Solutions

No solutions today! Otherwise I would just be giving away the answers 😅

📘 Connected Resources

  • The Organic Chemistry Tutor. Addition of Vectors By Means of Components - Physics. January 2021.
  • The Organic Chemistry Tutor. Cross Product of Two Vectors Explained! April 2017.
  • The Organic Chemistry Tutor. Dot Product of Two Vectors. May 2021.
  • The Organic Chemistry Tutor. Vector Operations - Adding and Subtracting Vectors. April 2023.
  • The Organic Chemistry Tutor. Vectors - Basic Introduction - Physics. January 2021.