Three-Dimensional Coordinate SystemA series of free Multivariable Calculus Video Lessons.
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.
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.
The Midpoint and Distance Formulas in 3D There are two formulas that are important to remember when considering vectors or positions in the 3D coordinate System. The midpoint formula and the distance formula in 3D. The midpoint and distance formula in 3D can be derived using a method of addition of the geometric representation of vectors. In order to understand the derivation of the distance formula in 3D we must understand 3D vector operations.
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.
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