# Dot Product

The **dot product **or **scalar product** of two vectors is the product of their magnitudes multiplied by the cosine of the angle that they form.

It can also be expressed as:

#### Example

Find the dot product of two vectors whose coordinates in an orthonormal basis are: (1, 1/2, 3) and (4, −4, 1).

(1, 1/2, 3) · (4, −4, 1) = 1 · 4 + (1/2) · (−4) + 3 · 1 = 4 −2 + 3 = 5

**Magnitude of a Vector**

#### Example

Find the magnitude of a vector with coordinates = (−3, 2, 5) in an orthonormal basis.

**Angle between Two Vectors**

#### Example

Determine the angle between the vectors = (1, 2, −3) and = (−2, 4, 1).

#### Orthogonal Vectors

Two vectors are orthogonal or perpendicular if their **dot product** is **zero**.

#### Example

Calculate the x and y values for the vector (x, y, 1) that is orthogonal to the vectors (3, 2, 0) and (2, 1, −1).

### Properties of the Dot Product

### 1Commutative

### 2 Associative

### 3 Distributive

### 4

#### Geometric Interpretation of the Dot Product

### Scalar Projection

### Vector Projection

The vector projection is the unit vector of by the scalar projection of u on v.

The scalar projection of u on v is the magnitude of the vector projection of u on v.

#### Exercise

Given the vectors and , calculate:

1. The magnitudes of and ·

2. The dot product of and ·

3. The angle.

4. The value of m for which vectors and are orthogonal.