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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:

dot product definition

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

Dot product calculation

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

dot product calculation

Example (calculation in two dimensions):

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

Solution:

solution


Example (calculation in three dimensions):

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

Solution:

solution

Calculating the Length of a Vector

The length of a vector vector A is: Length of a Vector

Example:

Vector A is given by vector A. Find |A|.

Solution:

solution

The angle between two vectors

The angle between two nonzero vectors A and B is

dot product calculation

Example: (angle between vectors in two dimensions):

Determine the angle between vector A and vector B.

Solution:

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

angle solution

The angle is,

angle solution


Example: (angle between vectors in three dimensions):

Determine the angle between vector A and vector B.

Solution:

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

angle solution

The angle is,

angle solution

Orthogonal vectors

If two vectors are orthogonal then: vectors.

Example:

Determine if the following vectors are orthogonal: orthogonal vectors

Solution:

The dot product is

orthogonal vectors

So, the two vectors are orthogonal.