Introduction
Some quantities—like temperature or mass—need only a number. But others, like velocity or force, require both a number and a direction. These are vectors.
Prerequisite Connection
You understand the coordinate plane and can calculate distance using the Pythagorean theorem.
Today's Increment
We define vectors, learn component notation , and calculate magnitude.
Why This Matters
Vectors are essential in physics (force, velocity), computer graphics (game engines), and multivariable calculus.
Vector Fundamentals
What is a Vector?
A vector is a mathematical object with both magnitude (length) and direction. We write it as or .
Scalar vs. Vector
Scalars are just numbers (temperature, mass). Vectors need direction too (velocity, force, displacement).
Component Form
The vector moves units horizontally and units vertically.
Magnitude (Length)
This comes directly from the Pythagorean theorem.
Position vs. Free Vectors
- • Position vector: Starts at the origin and ends at a point
- • Free vector: Can be moved anywhere; only magnitude and direction matter
Interactive: Explore a Vector
Worked Examples
Example 1: Finding Magnitude
Find the magnitude of .
Apply the magnitude formula
Solution
Example 2: Vector from Two Points
Find the vector from point to point .
Subtract coordinates
Find the magnitude
Solution
with magnitude 5
Example 3: Equal Vectors
Are vectors starting at the origin and from to equal?
Find vector v in component form
Compare components
Both vectors have the same components:
Solution
Yes, they are equal. Vectors with the same magnitude and direction are equal, regardless of where they start.
Common Pitfalls
Confusing points and vectors
A point is a location. A vector is a displacement. Note the different brackets!
Wrong direction for vector between points
The vector from A to B is . Subtracting in the wrong order gives the opposite direction.
Forgetting to square both components
For , you need , not .
Real-World Application
GPS Navigation
When your GPS tells you to travel "3 miles north, then 4 miles east," it's describing two vectors. The shortest path is the resultant vector with magnitude miles.
Aircraft pilots use vectors constantly: wind velocity is a vector, so is the plane's airspeed. The actual ground track combines these using vector addition.
Practice Quiz
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