Introduction
Sometimes we need to find how much of one vector lies "along" another. This is called projection. It answers the question: "What component of force is actually doing useful work?"
Prerequisite Connection
You know the dot product and can test for orthogonality.
Today's Increment
We learn scalar projection, vector projection, and the physics formula for work.
Why This Matters
Projection is fundamental in physics (work, energy), computer graphics (shadows), and machine learning.
Projections and Work
Scalar Projection (Component)
The signed length of along
Vector Projection
The actual vector that lies along
Work (Physics)
Work equals the dot product of force and displacement
Understanding Projection
- • Scalar projection: Can be positive or negative (direction matters)
- • Vector projection: Always points along (or opposite)
- • If vectors are perpendicular, projection is zero
- • If vectors align, projection equals the original vector
Interactive: Vector Projection onto Horizontal
Worked Examples
Example 1: Finding Projections
Find the scalar and vector projection of onto .
Step 1: Find dot products
Step 2: Scalar projection
Step 3: Vector projection
Solution
Scalar: 3, Vector:
Example 2: Calculating Work
A force (in Newtons) moves an object along displacement (in meters). Find the work done.
Apply work formula
Solution
Work = 40 Joules. Only the horizontal component of force (4 N) contributes to work along the horizontal path.
Example 3: Force on an Incline
A 50 lb force is applied at a 30° angle to pull a wagon along the ground. Find the effective horizontal force.
Step 1: Express force as a vector
Step 2: Find horizontal component
Project onto :
Solution
The effective horizontal pulling force is approximately 43.3 pounds.
Common Pitfalls
Confusing scalar and vector projection
Scalar projection is a number (divide by ). Vector projection is a vector (divide by and multiply by ).
Wrong direction in projection
is NOT the same as . The subscript indicates which vector we project onto.
Forgetting units in work problems
Work has units! Force (N) × distance (m) = Work (J). Don't forget to include the unit in your answer.
Real-World Application
Physics: Work and Energy
When pushing a lawnmower, you apply force at an angle. Only the horizontal component does work moving the mower forward. The vertical component just pushes down on the ground (wasted effort!).
That's why naturally extracts just the useful component. If you push perpendicular to motion, no work is done—even though you're exerting force!
Practice Quiz
Loading...