Vector Products and Vector Equations of Geometries - Introduction to Vectors
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MTH 104: VectorsIntroduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry.
Curated for first-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.
Introduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry. Curated for first-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.
Course Chapters
1Scalar Products
Scalar (dot) product of two vectors and its properties.
Chapter lessons
1.Introduction11:23
Meaning, types and need for vector products.
2.Definition3:34
Formal definition of the scalar or dot product of two vectors, and its relation to the projection of a vector on another vector.
3.Perpendicular vectors17:12
Scalar products of perpendicular vectors and unit vectors; scalar product of two vectors in terms of their mutually-perpendicular components.
4.Direction cosines24:18
Relationships between the scalar product, direction cosines of vectors and the angle between two vectors.
5.Properties16:14
Properties of the scalar product of two vectors.
6.Worked examples (1)17:00
Worked examples on the scalar product of two vectors.
7.Worked examples (2)20:42
More worked examples on the scalar product of two vectors.
8.Worked examples (3)8:24
More worked examples on the scalar product of two vectors.
9.Worked examples (4)19:59
More worked examples on the scalar product of two vectors.
10.Worked examples (5)37:04
More worked examples on the scalar product of two vectors.
2Vector Products
Vector (cross) product of two vectors and its properties.
Chapter lessons
1.Definition24:07
Definition, notations and direction of the vector or cross product of two vectors.
2.Parallel vectors37:00
Vector product of parallel and anti-parallel (like and unlike) vectors and unit vectors; vector product of two vectors in terms of their Cartesian components.
3.Collinearity11:20
Re-examining the collinearity of two vectors (three points) in the light of cross products.
4.Geometric meaning9:44
A geometric interpretation of the magnitude of a cross product of two vectors as the area of some parallelogram.
5.Properties9:33
Properties of the vector product of two vectors.
6.Worked examples (1)23:39
Worked examples on the vector product of two vectors and its implications.
7.Worked examples (2)32:57
More worked examples on the vector product of two vectors and its implications.
8.Worked examples (3)45:47
More worked examples on the vector product of two vectors and its implications.
9.Worked examples (4)29:40
More worked examples on the vector product of two vectors and its implications.
3Scalar Triple Products
Scalar triple product of three vectors and its properties.
Chapter lessons
1.Definition21:19
Formal definition of the scalar triple product and its definition in terms of Cartesian components.
2.Geometric meaning15:36
Geometric meaning of the scalar triple product of three vectors as the volume of some parallelepiped.
3.Linear dependence29:55
Examining the linear dependence (or coplanarity) of three vectors by their scalar triple product.
4.Properties12:58
Properties of the scalar triple product of three vectors.
5.Worked examples (1)22:03
Worked examples on the scalar triple product of three vectors.
6.Worked examples (2)29:35
More worked examples on the scalar triple product of three vectors.
7.Worked examples (3)17:51
More worked examples on the scalar triple product of three vectors.
8.Worked examples (4)20:59
More worked examples on the scalar triple product of three vectors.
4Vector Triple Products
Vector triple product of three vectors and its properties.
Chapter lessons
1.Definition12:31
Formal definition of the vector triple product or box product of three vectors.
2.Formula40:20
Simplifying the vector triple product using scalar coefficients.
3.Properties8:40
Properties of the vector triple product of three vectors.
4.Worked examples (1)37:37
Worked examples on vector triple products.
5.Worked examples (2)16:45
More worked examples on vector triple products.
6.Worked examples (3)30:43
More worked examples on vector triple products.
5Vector Equations
Solutions of vector equations with unknown vectors; solutions of vector equations with unknown scalars.
Chapter lessons
1.Techniques44:43
An overview of the solution techniques for vector equations with unknown vectors or unknown scalars.
2.Worked examples (1)49:33
Worked examples on vector equations with unknown vectors.
3.Worked examples (2)55:19
More worked examples on vector equations with unknown vectors.
4.Worked examples (3)37:43
More worked examples on vector equations with unknown vectors.
5.Worked examples (4)51:02
More worked examples on vector equations with unknown vectors.
6.Worked examples (5)26:05
More worked examples on vector equations with unknown vectors.
7.Worked examples (6)44:26
More worked examples on vector equations with unknown vectors.
8.Worked examples (7)56:22
Worked examples on vector equations with unknown scalars.
9.Worked examples (8)54:19
More worked examples on vector equations with unknown scalars.
6Vector Equations of a Line
Direction vector and vector equation of a straight line,; angle between two straight lines; intersecting lines, parallel lines and skew lines; shortest distance between two skew lines.
Chapter lessons
1.Introduction14:07
An introduction to the vector equations of geometries.
2.Direction and one point
Vector equation of a straight line through a given point in a given direction.
3.Two points
Vector equation of a straight line through two given points.
4.Worked examples (1)
Worked examples on the vector equation of a straight line.
5.Worked examples (2)
More worked examples on the vector equation of a straight line.
6.Distance from a point
Perpendicular distance from a point to a given straight line.
7.Angle between two lines
How to find the angle between two lines from their vector equations.
8.Intersection
Intersection of two lines; intersection of a line with a plane.
9.Skew lines
Meaning of skew lines and the perpendicular distance between two skew lines.
10.Worked examples (3)
More worked examples on the vector equation of a straight line.
11.Worked examples (4)
More worked examples on the vector equation of a straight line.
12.Worked examples (5)
More worked examples on the vector equation of a straight line.
13.Worked examples (6)
More worked examples on the vector equation of a straight line.
14.Worked examples (7)
More worked examples on the vector equation of a straight line.
7Vector Equations of a Plane
Vector equations of a plane; normal vector of a plane; parallel planes and the distance between them; angle between two planes; angle between a line and a plane.
Chapter lessons
1.A point and two lines
Vector equation of a plane through a given point and parallel to two given lines.
2.Three non-collinear points
Vector equation of a plane containing three given non-collinear points.
3.A point and a normal vector
Vector equation of a plane containing a given point and normal to a given vector (or line).
4.Worked examples (1)
Worked examples on vector equations of a plane.
5.Worked examples (2)
More worked examples on vector equations of a plane.
6.Worked examples (3)
More worked examples on vector equations of a plane.
7.Parallel planes
Parallel planes, their equations and minimum (perpendicular) distance apart.
8.Angle between two planes
How to find the angle between two intersecting planes.
9.Intersection with a line
Re-examining the point of intersection of a line and a plane; angle between a line and a plane.
10.Intersection of two planes
How to find the equation of the line of intersection of two planes.
11.Worked examples (4)
More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.
12.Worked examples (5)
More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.
13.Worked examples (6)
More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.
14.Worked examples (7)
More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.
15.Worked examples (8)
More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.
8Vector Equations of a Circle
Vector equations of a circle in x-y plane, a circle in space; non-parametric vector and standard forms; equation of the plane containing a circle.
Chapter lessons
1.x-y plane
Vector equation of a circle in x-y plane.
2.Three dimensions
Vector equation of a circle in a three-dimensional space.
3.Non-parametric equation
General non-parametric of equation of a circle in a plane; non-parametric of equation of a circle in the x-y plane.
4.Worked examples (1)
Worked examples on vector equations of a circle.
5.Worked examples (2)
More worked examples on vector equations of a circle.
6.Worked examples (3)
More worked examples on vector equations of a circle.
9Vector Equations of an Ellipse
Parametric and non-parametric vector equations of an ellipse.
Chapter lessons
1.x-y plane
Parametric equation of an ellipse in the x-y plane.
2.Three dimensions
Parametric equation of an ellipse in three dimensions.
3.Non-parametric equations
Non-parametric equations of an ellipse in two and three dimensions.
4.Worked examples (1)
Worked examples on parametric and non-parametric equations of an ellipse.
5.Worked examples (2)
More worked examples on parametric and non-parametric equations of an ellipse.
10Vector Equations of a Parabola
Parametric and non-parametric vector equations of a parabola.
Chapter lessons
1.x-y plane
Parametric vector equations of a parabola in the x-y plane.
2.Three dimensions
Parametric vector equations of a parabola in 3 dimensions.
3.Non-parametric equations
Non-parametric equations of a parabola in two and three dimensions.
4.Worked examples (1)
Worked examples on parametric and non-parametric equations of a parabola.
5.Worked examples (2)
More worked examples on parametric and non-parametric equations of a parabola.
11Vector Equations of a Hyperbola
Parametric and non-parametric vector equations of a hyperbola.
Chapter lessons
1.x-y plane
Parametric vector equations of a hyperbola in the x-y plane.
2.Three dimensions
Parametric equation of a hyperbola in three dimensions.
3.Non-parametric equations
Non-parametric equations of a hyperbola in two and three dimensions.
4.Worked examples (1)
Worked examples on the parametric and non-parametric equations of a hyperbola.
5.Worked examples (2)
More worked examples on the parametric and non-parametric equations of a hyperbola.