Lesson Description
This lesson introduces vectors in the coordinate plane. Students learn how vectors are represented, how to compute component form, and how to calculate vector magnitude using the distance formula. These ideas form the foundation for later vector operations and applications.
Learning Objectives
- Represent vectors using coordinate components.
- Determine vectors between two points.
- Calculate the magnitude of vectors.
- Interpret vectors geometrically in the coordinate plane.
Topics Covered
- Vector notation and directed segments
- Component form of vectors
- Vectors between two coordinate points
- Vector magnitude formula
- Geometric interpretation of vectors
Key Terms
- Vector: A quantity with both magnitude and direction.
- Component Form: Representation of a vector as \( \langle x, y \rangle \).
- Magnitude: Length of a vector.
- Initial Point: Starting point of a vector.
- Terminal Point: Ending point of a vector.
Lesson Examples
Example 1: Vector from Two Points
Find the vector from A(2, 3) to B(7, 9).
- Subtract coordinates: \( \langle 7-2,\; 9-3 \rangle \)
- Vector = \( \langle 5, 6 \rangle \)
Example 2: Magnitude of a Vector
Find the magnitude of \( \langle 5, 6 \rangle \).
- Use magnitude formula: \[ |\vec{v}| = \sqrt{5^2 + 6^2} \]
- \[ = \sqrt{25 + 36} = \sqrt{61} \]
Example 3: Vector in Standard Position
A vector from the origin to point (4, −3) has component form \( \langle 4, -3 \rangle \) and magnitude \( \sqrt{4^2 + (-3)^2} = 5 \).
Practice Problems
- Find the vector from (1, 2) to (6, 10).
- Find the magnitude of \( \langle 8, 15 \rangle \).
- Find the vector from (−3, 4) to (5, −2).
- Compute the magnitude of \( \langle -7, 24 \rangle \).
Tips & Common Mistakes
- Always subtract coordinates in correct order: terminal minus initial.
- Do not forget to square negative values correctly.
- Magnitude is always non-negative.
- Vectors describe movement, not location.
Summary
Vectors represent direction and magnitude in the coordinate plane. Using component form, we can easily compute vector length and prepare for vector operations such as addition, subtraction, and dot products in later lessons.
Challenge Problems
- Find points B such that vector AB equals \( \langle 6, -4 \rangle \) when A is (3, 2).
- Find a vector with magnitude 10 in the first quadrant.
- Show that vectors \( \langle 3, 4 \rangle \) and \( \langle -3, -4 \rangle \) have equal magnitudes.