Introduction to Vectors
A vector has both magnitude (how much) and direction (which way). Unlike plain numbers, vectors carry directional information — making them indispensable in physics and engineering.
Introduction
A scalar (like temperature or speed) is just a number. A vector adds direction: “30 mph northeast” vs. just “30 mph.” You'll learn to represent vectors geometrically (arrows) and algebraically (component form).
Past Knowledge
Coordinate plane, Pythagorean Theorem, basic trig ratios.
Today's Goal
Understand vector notation, magnitude, direction angle, and component form.
Future Success
Vector operations (7.9) and the dot product (7.10) build on these foundations.
Key Concepts
Component Form
A vector from origin to point is written:
Magnitude (Length)
Direction Angle
(Adjust for quadrant using reference angles.)
Unit Vector
A vector with magnitude 1:
Standard Unit Vectors
(horizontal) and (vertical). Any vector can be written as .
Worked Examples
Magnitude and Direction
Given: . Find magnitude and direction.
Answer: Magnitude = 5, Direction ≈ 53.1°
Component Form from Magnitude & Angle
Given: , direction .
Answer:
Finding the Unit Vector
Given: . Find its unit vector.
Answer:
Common Pitfalls
Wrong Quadrant for Direction Angle
only returns values in Q1 and Q4. If the vector is in Q2 or Q3, add 180° to get the correct direction angle.
Real-Life Applications
Navigation — Wind & Current
Pilots and sailors deal with vectors constantly. Wind speed + direction = a vector. Current speed + direction = another. The actual path of a plane or boat is the vector sum of its desired heading and these environmental forces.
Practice Quiz
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