A series of free Multivariable Calculus Video Lessons, including examples and solutions.
Introduction to the 3D Coordinate System
With vectors, we begin to work more with the 3D coordinate system. In the 3D coordinate system there is a third axis, and in equations there is a third variable. We will work with vectors in the 3D coordinate system and learn how to interpret the coordinates an of a vector in the 3D coordinate system. With the introduction to the 3D coordinate system, we also encounter other vector operations, lines and planes.
Distances and Midpoint Formula
Equation of a Sphere
Planes and Other Surfaces
Vectors in Space
Lines in Space
3D Vector Operations
Although they are similar to 2D vector operations, it is good to get practice doing 3D vector operations. 3D vector operations include addition and scalar multiplication, the dot product and the calculation of magnitude. The biggest difference in these 3D vector operations is an added step of computation. With 3D vector operations we can do computation such as find the angle between vectors in space.
Find the Sum of Scalar Multiples of Two Vectors in 3D (Component Form)
This example shows how to find the sum of scalar multiples of two vectors in space. The results are shown graphically
Dot Product of Vectors - 3D
This video provides several examples of how to determine the dot product of vectors in three dimensions and discusses the meaning of the dot product.
Lines in 3D
In the 3D coordinate system, lines can be described using vector equations or parametric equations. Lines in 3D have equations similar to lines in 2D, and can be found given two points on the line. In order to understand lines in 3D, one should understand how to parameterize a line in 2D and write the vector equation of a line. Perpendicular, parallel and skew lines are important cases that arise from lines in 3D.
This video explains how to determine the parametric equations of a line in 3D.
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