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Vector Sets for Describing Mathematical Shapes

Explore how sets of points as vectors are used to describe lines, curves, and surfaces in mathematics. Learn how vectors can express geometric elements like lines, parabolas, circles, and planes. Practice examples included.

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Vector Sets for Describing Mathematical Shapes

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  1. Chapter 3 Section 3.1 Introduction

  2. Sets of Points and Vectors Sets of points in can be written or expressed as sets of vectors in where the components of the vector satisfy certain conditions. In particular in and a set of vectors can be used to describe of lines, curves or surfaces and the reverse is also possible that a line, curve or surface can be described by a set of vectors. These are vectors in with and . A substitution gives: This is a parabola with vertex at the origin opening up. Example Describe and graph: These are vectors in with and . A substitution gives: This is a line with slope -⅓ and y-intercept 2. Example Describe and graph:

  3. These are vectors in with and . Subtracting 1 from y and squaring both sides: Circle with center and radius 2. Example Describe and graph: z Example Describe and graph: y These are vectors in . If we let this becomes the vector form of the line given above with point and direction vector x Example: Describe the points on the curve as a set of vectors.

  4. z Example Describe and graph: These are vectors in They have and . Substituting we get: This is a plane with point and normal vector y x Example Describe the points on the plane: as a set of vectors in Solve the equation for y. Let and be independent variables and a dependent variable.

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