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Dot product examples
The dot product
Introduction to the dot product with a focus on its basic geometric properties.
The formula for the dot product in terms of vector components
Derivation of the component formula for the dot product, starting with its geometric definition based on projection of vectors.
The cross product
Introduction to the cross product with a focus on its basic properties. Includes an interactive graphic to illustrate these properties of the cross product.
The formula for the cross product
Derivation of the formula to calculate the cross product of two vectors.
Cross product examples
Examples of calculating the cross product.
The scalar triple product
Definition of the scalar triple product and derivation of its formula. Include interactive graphic to illustrate its properties.
Scalar triple product example
Example showing how to calculate the scalar triple product.
The zero vector
The zero vector is the unique vector having zero length. The direction of the zero vector is undefined.
Multiplying matrices and vectors
How to multiply matrices with vectors and other matrices.
Matrix and vector multiplication examples
Examples demonstrating how to multiply matrices and vectors.
Vectors in arbitrary dimensions
A brief introduction to n-dimensional vectors.
The transpose of a matrix
Definition of the transpose of a matrix or a vector.
Dot product in matrix notation
How to view the dot product between two vectors as a product of matrices.
Examples of n-dimensional vectors
Examples showing the practical use of vectors in more than three dimensions.
An introduction to vectors
A introduction to the concept of a vector as an object with magnitude and direction.
Vectors in two- and three-dimensional Cartesian coordinates
A introduction to representing vectors using the standard Cartesian coordinate systems in the plane and in three-dimensional space.
The derivative of a power function
Using the limit definition of the derivative to calculate the derivative of a power function.
Dot product examples