Culminating Review for Vectors

Size: px

Start display at page:

Download "Culminating Review for Vectors"

Shanna Pope
5 years ago
Views:

1 Culminating Review for Vectors An Introduction to Vectors Applications of Vectors Equations of Lines and Planes 4 12 Relationships between Points, Lines and Planes

2 An Introduction to Vectors Geometric Vectors A geometric vector is a representation of a vector using an arrow diagram, or directed line segment, that shows both magnitude and direction. B Referred to as or A 5 km 30 o Magnitude is or

3 Directions may be represented as a bearing; which is a compass measurement where the angle is measured from the north in a clockwise direction. quadrant bearing; which is a measurement between 0 o and 90 o east or west from the north-south line. N N 30 o 50 o N60 o E or 060 o S40 o W or 220 o

4 Cartesian (Algebraic) Vectors We identify as a Cartesian vector because its endpoints can be defined using Cartesian coordinates. position vector In 3-space

5 Operations with Vectors Scalar Multiplication Geometric Vectors Algebraic Vectors If If Then Then

6 Vector Addition Geometric Vectors Algebraic Vectors If and If and

7 Vector Subtraction Geometric Vectors Algebraic Vectors If and If and

8 Applications of Vectors The Dot Product Geometric Vectors Algebraic Vectors If and We can use the dot product to solve for the angle between any two vectors:

9 The Cross Product Algebraic Vectors If and Geometric Vectors OR a 2 a 3 a 1 a 2 b 3 b 1 b 2 x y z b 2 Note: Direction is determined using the right-hand rule down product minus the up product The magnitude of the cross product is the area of a parallelogram defined by and

10 Scalar and Vector Projections The scalar projection of onto is The vector projection of onto is Recall:

11 Equations of Lines and Planes Equations of Lines Vector Equation: If you equate the respective x, y and z components we obtain the parametric equations: x = x o + ta y = y o + tb z = z o + tc

12 x = x o + ta y = y o + tb z = z o + tc If you isolate each parametric equation for the parameter, t, we have Combining these statements gives the symmetric equation of a line:

13 Equations of Planes Vector Equation: [x, y, z] = [x o, y o, z o ] + s[a 1, a 2, a 3 ] + t[b 1, b 2, b 3 ] If you equate the respective x, y and z components we obtain the parametric equations: x = x o + sa 1 + tb 1 y = y o + sa 2 + tb 2 z = z o + sa 3 + tb 3

14 Cartesian Equation: The Cartesian (or scalar) equation of a plane is of the form Ax + By + Cz + D = 0 with normal n = A, B, C. The normal n is a vector perpendicular to all vectors in the plane. Given two directions vectors the normal to a plane is determined by calculating the cross product of the direction vectors.

15 Relationships between Points, Lines and Planes The Intersection of Two Lines l 1 : [x, y, z] = [ 3, 1, 4] + s[ 1, 1, 4] l 2 : [x, y, z] = [1, 4, 6] + t[ 6, 1, 6] Convert to parametric form l 1 l 2 x = 3 s x = 1 6t y = 1 + s z = 4 + 4s y = 4 t z = 6 + 6t Select any two of the three equations and equate them 3 s = 1 6t 1 + s = 4 t Solve for s and t and sub these results into both lines to check your solution

16 The Intersection of a Line and a Plane l : [x, y, z] = [3, 1, 2] + s[1, 4, 8] : 4x + 2y z 8 = 0 Convert to line to parametric form x = 3 + s y = 1 4s Equate z = 2 8s 4(3 + s) + 2(1 4s) (2 8s) 8 = 0 Solve for s sub this result into the line to determine the point of intersection

17 The Intersection of Two Planes 1 : x y + z = 3 2 : 2x + y 2z = 3 Analyze the normals; what do they tell you? Eliminate one of the variables, for example, z, gives 4x y = 9 Isolate for a variable y = 4x 9 Introduce a parameter Let x = t y = 4t 9 Substitute x = t and y = 4t 9 into one of the planes t (4t 9) + z = 3 z = 3t 6 The line of intersection is x = t y = 4t 9 z = 3t 6

18 The Intersection of Three Planes 1 : 2x + y z = 3 2 : x y + 2z = 0 3 : 3x + 2y z = 5 Create two new equations 2 1 : 4x + 2y 2z = 6 2 : x y + 2z = : 5x + y = 6 1 : 2x + y z = 3 3 : 3x + 2y z = : x y = 2 Solve for x and y, and use these values and one of the planes to determine the value of z. Verify the results using the other planes.

19 Culminating Review for Calculus Limits Derivatives Velocity and Acceleration Extreme Values Optimization Curve Sketching 4 12

20 Limits lim x a f x = L Tells us that the value of f x approaches L as x approaches the value a. (a) (b) (c) lim x 1 lim x 1 + f x = f x = lim f x = x 1 (d) f 1 =

21 Algebraic methods lim x 0 3x2 2x + 7 direct substitution lim x 4 x 2 + 7x + 12 x + 4 factor and simplify lim x 0 x x multiply by the conjugate x or use substitution, let u = x + 9 lim x 0 3 x x use substitution, let u = 3 x + 27

22 Continuity A function f is continuous at x = a if 1. lim x a f(x) exists 2. lim x a f x = f a

23 Derivatives Derivative of a function using first principles: f x = lim h 0 f x + h f x h Power Rule f x = x n f x = nx n 1 Power of a Function Rule f x = g x n f x = n g x n 1 g x

24 Product Rule f x = p x q x f x = p x q x + p x q x Quotient Rule p x f x = q x f x = p x q x p x q x q x 2 Chain Rule f x = g h x f x = g h x h x

25 Exponential Rules f x = e x f x = e x g x f x = e f x = e g x g x f x = b x f x = b x ln b f x = bg x f x = b g x (ln b) g x

26 Trigonometric Rules f x = sin x f x = cos x f x = cos x f x = sin x f x = sin g x f x = cos g x g x f x = cos g x f x = sin g x g x

27 Velocity and Acceleration Position, velocity and acceleration Given position as s t then Velocity is v t = s t Acceleration is a t = v t = s t

28 Extreme Values Algorithm for Finding Maximum and Minimum (Extreme) Values: If a function f(x) has a derivative at every point in the interval a x b, calculate f(x) at: all points in the interval a x b, where f x = 0 (critical values) the endpoint x = a and x = b The maximum value of f(x) on the interval a x b, is the largest of these values, and the minimum value of f(x) on the interval is the smallest of these values

29 Optimization An Algorithm for Solving Optimization Problems Identify the variables. Determine a function that represents the quantity to be optimized. Determine the domain of the function to be optimized. Use the algorithm for extreme values to find the absolute maximum or minimum in the domain. Conclude.

30 Curve Sketching An Algorithm for Curve Sketching Using f x Determine any intercepts. Determine any vertical asymptotes and determine the behaviour on either side of the asymptote. Determine any horizontal or oblique asymptotes. Identify if the function values approach the asymptote from above or below. Using f x Determine any critical numbers by finding where f x = 0 or where f x is undefined. Determine the intervals of increase/decrease, and identify any local or absolute extrema.

31 Using f x Determine possible points of inflection by finding where f x = 0 or where f (x) is undefined. Determine the intervals of concavity, and identify any points of inflection. Sketch the curve.

and Calculus and Vectors

and Calculus and Vectors Autograph is spectacular dynamic software from the UK that allows teachers to visualise many of the mathematical topics that occur in the Ontario Grade 12 CALCULUS and VECTORS