A two-vector or bivector is a tensor of type and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then where the f α β are the components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition. A bivector may operate on a one-form, yielding a vector: , although a problem might be which of the upper indices of the bivector to contract with. (This problem does not arise with mixed tensors because only one of such tensor's indices is upper.) However, if the bivector is symmetric then the choice of index to contract with is indifferent. An example of a bivector is the stress–energy tensor. Another one is the orthogonal complement of the metric tensor. (Wikipedia).
Ex: Find Two Unit Vectors Orthogonal to Two Given Vectors
This video explains how to use the cross product of two vectors to determine two unit vectors that are orthogonal to two given vectors. http://mathispower4u.com
From playlist Vectors in Space (3D)
Vector Space of All 2 by 2 Matrices (M22): Find the Zero Vector, Mult Identity and Additive Inverse
This video explains how to find the zero vector , the multiplicative, and additive inverse of the vector space of all 2 by 2 matrices.
From playlist Vector Spaces
How to determine if two vectors are parallel, orthogonal or neither
Learn how to determine if two vectors are orthogonal, parallel or neither. You can setermine whether two vectors are parallel, orthogonal, or neither uxsing the dot/cross product or using the slope formula. Recall that two lines are parallel if they both have the same slope and are perpem
From playlist Vectors
This video explains how to determine if vectors are parallel. http://mathispower4u.yolasite.com/
From playlist Vectors
Systems of Equations with Elimination Two Variables Two Equations Example 2
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Systems of Equations with Elimination Two Variables Two Equations Example 2
From playlist Systems of Equations
Definition Check: One-to-One and Onto Functions
This video reinforces the definitions of one-to-one functions and onto functions.
From playlist One-to-One and Onto Transformations
Systems of Equations with Elimination Two Variables Two Equations Example 1
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Systems of Equations with Elimination Two Variables Two Equations Example 1
From playlist Systems of Equations
What is the dot product of two vectors? How is it useful? Free ebook https://bookboon.com/en/introduction-to-vectors-ebook (updated link) Test your understanding via a short quiz http://goo.gl/forms/2SGI5Kvpk9
From playlist Introduction to Vectors
What is the cross product of two vectors? How is it useful? Free ebook https://bookboon.com/en/introduction-to-vectors-ebook (updated link) Test your understanding via a short quiz http://goo.gl/forms/Ii3hPtwksX
From playlist Introduction to Vectors
This calculus 3 video tutorial provides a basic introduction into vectors. It contains plenty of examples and practice problems. My Website: https://www.video-tutor.net Patreon Donations: https://www.patreon.com/MathScienceTutor Amazon Store: https://www.amazon.com/shop/theorganicch
From playlist New Calculus Video Playlist
Lecture 2 | Modern Physics: Quantum Mechanics (Stanford)
Lecture 2 of Leonard Susskind's Modern Physics course concentrating on Quantum Mechanics. Recorded January 21, 2008 at Stanford University. This Stanford Continuing Studies course is the second of a six-quarter sequence of classes exploring the essential theoretical foundations of mode
From playlist Quantum Mechanics Prof. Susskind & Feynman
A Swift Introduction to Geometric Algebra
This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realiz
From playlist Miscellaneous Math
QED Prerequisites Geometric Algebra 6 - Multivector Products
This lesson begins by clearing up the definition of a "k-blade". Then we proceed to write down explicit expressions for most of the multivector products between vectors and bivectors, vectors and trivectors, and two bivectors. We do this using the "relative" formalism with a basis set give
From playlist QED- Prerequisite Topics
WildLinAlg2: Geometry with vectors
Please watch: "UNSWTV: Entertaining your curiosity" https://www.youtube.com/watch?v=bQ7UO8nxiL0 -~-~~-~~~-~~-~- We discuss basic constructions with vectors, and the laws that they satisfy. Affine combinations describe points lying on a line. We give applications to various elementary the
From playlist A first course in Linear Algebra - N J Wildberger
Basic Operations Geometric Vectors (Many Examples)
I introduce the Basic Operations of Geometric Vectors, addition of vectors, subtraction of vectors, scalar multiplication. Diagraming the addition of vectors using the Parallelogram Law 0:58 Triangular Method 3:31 19:30 Properties of Vectors Addition explained with sketches 7:10
From playlist Vector Math Lessons
Introduction to Linear Combinations and Span of Vectors
This video introduces and defines linear combinations and span of vectors.
From playlist Spanning Sets and Subspaces
Lecture 3: Exterior Algebra (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Geometric Algebra 19 3D Geometric Algebra
In this lesson we examine the 3D geometric algebra for the Euclidean case. This is simpler than the spacetime algebra we have been considering so far. First, we construct the basis for the 3D algebra and then begin to make contact with the standard vector algebra of elementary physics. Aft
From playlist QED- Prerequisite Topics
Cross Product Vector Product Example & Properties
I introduce the Vector Product, Cross Product, formula and show you how to write the formula by finding the determinant of a 3x3 matrix through the determinant of multiple 2x2 matrices. Example 1) Find the cross product of two vectors UxV 7:49 Example 2) Dot product to determine the rela
From playlist Vector Math Lessons
Ex: Find the Difference of Two Vectors Given in Linear Combination Form
This video explains how to find the difference of two vectors given as linear combinations of unit vectors. Site: http://mathispower4u.com
From playlist Vectors in 2D