Introduction to Vectors

🔖 Topics

• Properties of Vectors
• Vector Arithmetic
• Dot Product
• Cross Product
• Unit Vectors
• Normalization

🎯 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

🖥️ Animations, Simulations, Activities

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📝 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 $\vec{b}\times \vec{c}$?

✅ Partial Solutions

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📘 Connected Resources

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