# 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

#### Example

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

#### 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).

### 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.