The Dot Product
The dot product takes two vectors and returns a scalar. It measures how aligned two vectors are and lets you find the angle between them.
Introduction
Addition combines vectors into a resultant. The dot product does something entirely different: it multiplies two vectors to produce a single number that encodes the angle between them.
Past Knowledge
Vector components (7.8), magnitude, vector operations (7.9), inverse cosine.
Today's Goal
Compute dot products, find angles between vectors, and test for orthogonality.
Future Success
The dot product is foundational for projections, work calculations, and linear algebra.
Key Concepts
Component Formula
Geometric Formula
where is the angle between the vectors.
Finding the Angle
Orthogonality Test
Two vectors are perpendicular (orthogonal) if and only if . This is the fastest way to check for right angles.
What the Sign Tells You
| Dot Product | Angle | Meaning |
|---|---|---|
| Positive | Acute — vectors point roughly the same way | |
| Zero | Perpendicular | |
| Negative | Obtuse — vectors point roughly opposite |
Worked Examples
Computing a Dot Product
Given: , .
Answer: 2 (positive → acute angle between them)
Finding the Angle
Given: , . Find the angle.
,
Answer: (obtuse, confirmed by negative dot product)
Orthogonality Check
Are and perpendicular?
Answer: Yes — dot product is 0, so the vectors are orthogonal.
Common Pitfalls
Expecting the Dot Product to Be a Vector
The dot product is a scalar (a number), not a vector. Don't put angle brackets around your answer.
Real-Life Applications
Physics — Work
Work is defined as — the dot product of force and displacement. When you push a box at an angle, only the component of force parallel to the motion does work. The dot product captures exactly this.
Practice Quiz
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