Section 11.3

The Dot Product

Multiplying vectors to find angles and projections.

Introduction

The dot product (or scalar product) takes two vectors and returns a single number. While it looks like simple multiplication, it holds the secret to measuring how parallel two vectors are. If the dot product is zero, the vectors are perpendicular (orthogonal).

Interactive: Vector Projection

The green vector is the projection (shadow) of the blue vector onto the grey axis.

Key Formulas

Algebraic Definition

Sum of products of components:

Geometric Definition

Using the angle :

Orthogonal Test

Vectors are perpendicular if:

Vector Projection

Shadow of onto :

Worked Examples

Example 1: Dot Product & Orthogonality (Level 1)

Calculate for and . Are they orthogonal?

Step 1: Multiply Components

Step 2: Sum and Check

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Since the dot product is zero, the vectors are orthogonal (perpendicular).

Example 2: Angle Between Vectors (Level 2)

Find the angle between and .

1. Dot Product:

2. Magnitudes:

3. Cosine Formula:

Notice the angle is close to 90°, consistent with the dot product (1) being close to 0.

Example 3: Vector Projection (Level 3)

Find the projection of onto .

Warning: Remember the formula projects B onto A. The result is parallel to A.

Step 1: Numerator (Dot Product)

Step 2: Denominator (Magnitude Squared of A)

Step 3: Scale Vector A