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 « Basic Operations with Vectors
Linear Algebra - Vectors: (lesson 2 of 3)

## Dot Product

Definition:

The dot product (also called the inner product or scalar product) of two vectors is defined as:

Where |A| and |B| represents the magnitudes of vectors A and B and is the angle between vectors A and B.

### Dot product calculation

The dot or scalar product of vectors and can be written as:

Example (calculation in two dimensions):

Vectors A and B are given by and . Find the dot product of the two vectors.

Solution:

Example (calculation in three dimensions):

Vectors A and B are given by and . Find the dot product of the two vectors.

Solution:

### Calculating the Length of a Vector

The length of a vector is:

Example:

Vector A is given by . Find |A|.

Solution:

### The angle between two vectors

The angle between two nonzero vectors A and B is

Example: (angle between vectors in two dimensions):

Determine the angle between and .

Solution:

We will need the magnitudes of each vector as well as the dot product.

The angle is,

Example: (angle between vectors in three dimensions):

Determine the angle between and .

Solution:

Again, we need the magnitudes as well as the dot product.

The angle is,

### Orthogonal vectors

If two vectors are orthogonal then: .

Example:

Determine if the following vectors are orthogonal:

Solution:

The dot product is

So, the two vectors are orthogonal.