Lesson 14.4

The Dot Product and Orthogonality

A multiplication that produces a scalar—revealing the angle between vectors.

Introduction

The dot product (or scalar product) combines two vectors and produces a single number. This number encodes the angle between them—including whether they're perpendicular.

1

Prerequisite Connection

You know unit vectors and can express vectors in component form.

2

Today's Increment

We learn the dot product formula, how to find angles, and the test for orthogonality.

3

Why This Matters

The dot product underlies physics (work), graphics (lighting), and machine learning (similarity).

The Dot Product

Algebraic Definition

Multiply corresponding components, then add

Geometric Definition

Where is the angle between the vectors

Angle Between Vectors

Orthogonality Test

Two vectors are orthogonal (perpendicular) if and only if

Key Properties

  • • : Acute angle (less than 90°)
  • • : Right angle (exactly 90°)
  • • : Obtuse angle (greater than 90°)
  • • (dot product with itself gives magnitude squared)

Interactive: Explore the Dot Product

u · v = 6θ ≈ 53.1°

Worked Examples

Example 1: Computing a Dot Product

Find where and .

Apply the formula

Solution

(negative means obtuse angle)

Example 2: Finding the Angle Between Vectors

Find the angle between and .

Step 1: Compute the dot product

Step 2: Find magnitudes

Step 3: Apply angle formula

Solution

Example 3: Testing for Orthogonality

Are and orthogonal?

Compute the dot product

Solution

Yes, they are orthogonal! The dot product equals zero, meaning the vectors are perpendicular.

Common Pitfalls

Expecting a vector result

The dot product gives a scalar (number), not a vector. If you get a vector, you've made an error.

Forgetting to use arccos

The formula gives , not . You must apply to find the angle.

Confusing orthogonal with zero vector

means perpendicular. It does NOT mean either vector is the zero vector.

Real-World Application

3D Graphics: Lighting Calculations

When light hits a surface, how bright it appears depends on the angle. If is the direction to the light and is the surface normal, the brightness is proportional to .

When vectors align (light hits straight-on), the dot product is maximum. When perpendicular (light grazes the surface), it's zero. This is the core of computer graphics lighting models.

Practice Quiz

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