« Basic Operations with Vectors

Cross Product »

Definition:

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.