Lesson Description
This lesson introduces vector addition and subtraction in the coordinate plane. Students learn how vectors combine, how to compute resultant vectors, and how vectors model real-world displacement and motion problems.
Learning Objectives
- Add vectors using component form.
- Subtract vectors correctly.
- Interpret resultant vectors geometrically.
- Apply vector operations to word problems.
Topics Covered
- Vector addition using components
- Head-to-tail geometric interpretation
- Vector subtraction
- Resultant vectors
- Applied displacement problems
Key Terms
- Vector Addition: Combining vectors component-wise.
- Vector Subtraction: Adding the opposite vector.
- Resultant Vector: Final vector after combining vectors.
- Displacement: Change in position represented by a vector.
Lesson Examples
Example 1: Adding Vectors
Add \( \langle 3, 4 \rangle + \langle 2, -1 \rangle \).
- Add components: \( (3+2,\;4+(-1)) \)
- Result: \( \langle 5, 3 \rangle \)
Example 2: Subtracting Vectors
Find \( \langle 6, 2 \rangle - \langle 1, 5 \rangle \).
- Subtract components: \( (6-1,\;2-5) \)
- Result: \( \langle 5, -3 \rangle \)
Example 3: Applied Motion Problem
A person walks 4 units east and 3 units north, then 2 units west.
- First movement: \( \langle 4,3 \rangle \)
- Second movement: \( \langle -2,0 \rangle \)
- Total displacement: \( \langle 2,3 \rangle \)
Practice Problems
- Add \( \langle 7, -2 \rangle + \langle -3, 6 \rangle \).
- Find \( \langle 5, 9 \rangle - \langle 2, 4 \rangle \).
- A drone moves \( \langle 10,5 \rangle \) then \( \langle -6,3 \rangle \). Find final displacement.
- Add vectors \( \langle -4,7 \rangle + \langle 3,-2 \rangle \).
Tips & Common Mistakes
- Add or subtract components separately.
- Subtraction means adding the opposite vector.
- Check signs carefully when subtracting.
- Visual diagrams help avoid mistakes.
Summary
Vector addition and subtraction allow us to model combined motion and displacement. These operations form the foundation for physics, engineering, and navigation applications.
Challenge Problems
- Find two vectors whose sum equals \( \langle 8,2 \rangle \).
- A boat moves north 6 units and east 8 units. Find displacement magnitude.
- Find a vector that cancels \( \langle 5,-9 \rangle \).