Vector Equations IB Mathematics

Today Vector Equations of Lines Two and Three Dimensions Next week: Parameters, Parametric Equations The Algebra of Vector Equations Intersecting, Parallel and Coincident Lines

Why Do We Study This? Right now, you have one way to describe an object’s position: Cartesian coordinates (x, y) or (x, y, z) Vectors give us another way to describe an object’s position. If you pursue a career in engineering, aeronautics, physics or a number of other math- related fields, you will address problems that require a different way of viewing the position of objects…

Geometric Representation (see figure) The vector equation should describe the position of any point P on the line Path: Move from O to A Move parallel to vector AB to point P (from A to P)

Definitions, Terminology (see figure) The position vector for any point on the line is called r The position vector for the point through which a line passes is a The vector that is parallel to the line is b “lambda” is called the parameter

Vector Equations Defined If a line passes through the point with a position vector a, and is parallel to the vector b, then its equation is given below. “lambda” represents any real number, so that any point on the line is some multiple of b away from a fixed point

Determining a Vector Equation You must be able to: Determine the point through which it passes For example: (6,3) Determine a vector that is parallel to the line For example: -4i + 2j Note: There are multiple solutions for this…

Things to think about… Equations can be written in component form But…it is usually more convenient to use column vectors Equations may be manipulated by distributing lambda (or otherwise) But…it is usually more convenient to maintain standard form

Column vs. Component Forms

Skills Generate points on a line by using different values for the parameter Given a point and a parallel vector, determine the vector equation Given two points, determine the vector equation

Activity/Homework Exercise (a, c, e) 3 (a, c) Exercise (a, c)

Vector Equations 2 IB Mathematics

Today Homework Review Parameters, Parametric Equations, Cartesian Form Wednesday: The Algebra of Vector Equations Intersecting, Parallel and Coincident Lines When Objects Collide (starring Bruce Willis)

Parametric Equations X and Y vary independently according to a parameter In the case of vector equations, the standard parameter is lambda, but it can be represented by other symbols as well (the most common is T)

Cartesian Form Any given point on a line has a specific value for the parameter. With the vector equation in parametric form, lambda can be isolated to yield an equation in terms of x and y only This is Cartesian form

The Different Forms of Lines Slope-intercept (closely related to cartesian) Vector Parameter Cartesian

Homework Exercise (page 450) 4, 5, 7 (first and last on each)

Vector Equations 3 IB Mathematics

Today Homework Questions Angles between lines Finding Intersections Classifying Intersections Next class: more classifying intersections, When Objects Collide, other applications

Angles Between Lines To find the angle between two lines, find the angle between the vectors that describe their directions Use the Dot Product

Determining the Dot Product from components If you do not have the angle between two vectors, you can still obtain the dot product from the components of the vectors

Special Cases If two vectors are perpendicular, then the dot product is zero If two vectors are parallel, the vectors are a multiple of each other (b 1 =mb 2 where m is not zero) Or: the dot product is the product of the magnitudes.

Practice For the two pairs of lines on the right, find the acute angle that each pair forms

Intersections of Lines Set the equations equal to each other Solve the simultaneous equations for the parameter Substitute the parameter to get the position vector for the intersection

Practice Exercise (page 451) 9, 11, 17 Exercise (page 463) 13 (a, b only)

Vector Equations 4: Applications IB Mathematics

Today Homework Questions Classification Relationships Between Vectors in three dimensions Applications of Vector Equations Motion of an object (parameters) Objects colliding or hitting/missing a point

Problem of the Day For each pair, describe the relationship between the two lines. If possible, give the acute angle that the two lines form

Classifying Relationships Between Vectors Intersecting (meet at one point) “Perpendicular” is a special case Parallel (same plane, but do not meet) Coincident Skew

Perpendicular and Parallel If two vectors are perpendicular, then the dot product is zero If two vectors are parallel, the vectors are a multiple of each other (b 1 =mb 2 where m is not zero) Or: the dot product is the product of the magnitudes.

Coincident If you believe that two lines are parallel (b 1 =mb 2 where m is not zero), test if they share a common point If they share a point, they are coincident

Skew Lines For lines to be SKEW They exist in 3-space They do not intersect They are NOT parallel Concept: be aware that, in three dimensions, lines that do not intersect may or may not be parallel

Challenge, Part 1 SHOW that the two lines on the right are skew That is, show that they: are not parallel do not intersect at a point

Challenge, Part 2 Create two vector equations for lines that are skew and show that they are indeed skew

Application: Path of an Object A common application of vector equations is position over time. Position Initial position Velocity The parameter (t) Speed

Example

Practice Activity The positions of two bumper cars A and B at t seconds are described by the vector equations on the right Determine each car’s position after 3 seconds When will they collide (if at all)?

When Objects Collide If two objects collide, the horizontal (x) positions should be equivalent at the same time that the vertical (y) positions are equivalent Steps: Set the equations equal to one another Solve for t according the both x and y If the two values for t are equivalent, that is the point at which the objects collide

Practice 2 Exercise (page 455) 1 and 2

Challenge: Vector Fields Activity A vector field is a plane that is a set of vectors whose values are determined by a function At each point P, use the function to draw a vector whose starting point is P

Vector Test Next Week (Wed.) Material on the Previous Quiz Vector Equations of Lines (2-D and 3-D) Generating points, determining the equation, parametric equations Relationships between lines Angles, perpendicular and parallel lines, coincident lines, awareness of skew lines in 3-D Finding Intersections Applications (motion, collisions)

Material from the quiz Vector vs. Scalar Quantities Vectors in Component form and as a Column 2-D and 3-D vectors Position Vectors, Negative Vectors, the Zero Vector Magnitude, Unit Vectors Algebra, Operations with Vectors The Dot Product

Review Review the Quiz and Homework Problems Other problems in Revision Set C (starts on page 465) 1, 3, 8, 12, 14, 15, 16, 17