VQ - Modeling with Vector Quantities Module Overview

Modeling with Vector Quantities Module Overview

Introduction

In this module, we will learn about vectors and parametric equations! You may have heard of vectors before in Physics, but the big idea will be to understand how two forces work on object. You will utilize right triangle trigonometry as well as Unit Circle trigonometry.

A curve in the plane consists of a collection of points and the defining parametric equations. What is different here from your previous experience is that two distinct curves can have the same graph because the curve is traced out in different ways on the graph. Vector-valued functions incorporate parametric equations as component functions of the vector function. Likewise with polar curves, parametric equations serve as the vehicle for finding tangents to polar curves.

Essential Questions

How are vector and scalar quantities similar and different?
How can I use vector operations to model, solve and interpret real-world problems?
How can I represent addition, subtraction, and scalar multiplication geometrically?
What are different ways to geometrically represent the addition of vectors?
In what ways can matrices transform vectors?
What are some advantages of using parametric equations for graphing?
How can technology help when investigating graphs of parametric and polar curves?

Vectors Key Terms

The following key terms will help you understand the content in this module.

Vector – A mathematical object that has both magnitude and direction, expressed as v, equation image indicator $LaTeX: \overline{v}$ $\overline{v}$ , or〈a,b〉or as a directed line segment.

Scalar – A real number. A scalar has a magnitude but not direction.

Initial Point – The point at the tail of the arrow representing a vector. The initial point is the origin if the vector is in standard position.

Terminal Point – The point at the "tip" of the arrow representing a vector.

Magnitude of a Vector – The distance between vector's initial and terminal points denoted ‖v‖, | equation image indicator $LaTeX: \overline{v}$ $\overline{v}$ |, or |v|. $LaTeX: \lVert{v}\rVert = \lVert{a, b}\rVert =\sqrt{a^2 +b^2}$ $\lVert{v}\rVert = \lVert{a, b}\rVert =\sqrt{a^2 +b^2}$ .

Components of a Vector – a and b in the vector〈a,b〉. The horizontal component is a and the vertical component is b.

Parallel Vectors – Two or more vectors whose directions are the same or opposite.

Equivalent Vectors – Two or more vectors that have the same direction and magnitude.

Resultant Vector – The vector that results from adding two or more vectors.

Eliminating the parameter - Finding a rectangular equation that represents the graph of a set of parametric equation.

Parameter - An arbitrary constant whose value affects the specific nature but not the formal properties of a mathematical expression.

Parametric equations - A set of equations expressing a number of quantities as explicit functions of the same set of independent variables (parameters) and equivalent to some direct functional relationship between these quantities.

Plane curve - The parametric equations x = f(t) and y = g(t), where f and g are continuous functions of t, taken together with the graph of the parametric equations.

Polar axis - A horizontal ray beginning at the pole and extending to the right.

Polar coordinates - A pair of coordinates (r, ϴ) that locates a point in a plane by its distance r from the origin/pole and its angle of inclination theta ϴ.

Polar graph - The set of points whose polar coordinates r and ϴ satisfy the equation r = f(ϴ).

Position vector - A vector represented by a directed line segment whose initial point is the origin.

Smooth curve - A curve with parametric representation x = f(t) and y = g(t) on an interval I where f and g have continuous first derivatives on I and are not simultaneously 0, except possibly at the endpoints of I.

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