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Learn about the algebraic properties of vectors and how to write equations in parametric form. Understand vector operations and their applications in various graphs.
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Lesson 12 – 8 Algebraic Vectors & Parametric Equations Pre-calculus
Learning Objective • To write equations of vectors and parametric equations
In 12–7, we focused on the geometric aspect of vectors. 12–8 focuses on the algebraic properties Vectors & Parametric Equations (x, y) y x Note: the book will use ( , ) for a point AND a vector! Be careful! We will use for vectors is the norm/magnitude
Unit Vector: vector with magnitude 1 (horizontal) Vectors & Parametric Equations (vertical) is the unit vector with the same direction as Component Form: Can be expressed as Polar Form:
We can determine a vector if we know its initial and terminal points Vectors & Parametric Equations or 1. Given with initial point (2, 3) & terminal point (7, 9), determine the component form
2. Given with initial point (2, 3) & terminal point (7, 9), determine the polar form Vectors & Parametric Equations c 3. Given with initial point (2, 3) & terminal point (7, 9), determine the unit vector in the same direction as
Vector operations: If and Vectors & Parametric Equations Vector sum: Vector difference: Scalar Multiplication: 4 and. Find such that
In this picture, is a point and is a point. P(a, b) Vectors & Parametric Equations If you wanted to get to from , we could add a vector Q(x, y) Since we don’t know the size of the vector, we can multiply by a scalar to get to the point. Vector Equation of line: Now, The direction vector will be given to you, or you can find it by subtracting the 2nd point – 1st point that they give you.
In this picture, is a point and is a point. P(a, b) Vectors & Parametric Equations Q(x, y) So, This leads to and These two equations are called parametric equations with parameter of the line
5. Determine a direction vector of the line containing the two points and . Then find the equation of the line & a pair of parametric equations of the line. Vectors & Parametric Equations Direction vector Vector Equation of line: Parametric Equations: and We can also represent other graphs (not just lines) in parametric.
6. Graph the curve with parametric equations and . Find an equation of the curve that contains no other variables but & . Vectors & Parametric Equations square put together 5 -3 3 Ellipse -5
Assignment Pg. 651 #1, 3, 6, 9, 11, 15–18, 20, 21, 24, 29, 31, 33, 35