VECTORS APPLICATIONS NHAA/IMK/UNIMAP.

Slides:

Advertisements

Similar presentations

Vector Equations in Space Accelerated Math 3. Vector r is the position vector to a variable point P(x,y,z) on the line. Point P o =(5,11,13) is a fixed.

Advertisements

Euclidean m-Space & Linear Equations Euclidean m-space.

Parallel and Perpendicular Lines

Even More Vectors.

MA Day 9 – January 17, 2013 Review: Equations of lines, Section 9.5 Section 9.5 –Planes.

IB HL Adrian Sparrow Vectors: the cross product.

Lines and Planes in Space

11 Analytic Geometry in Three Dimensions

Vectors: planes. The plane Normal equation of the plane.

Section 9.5: Equations of Lines and Planes

Assigned work: pg. 449 #1abc, 2abc, 5, 7-9,12,15 Write out what “you think” the vector, parametric a symmetric equations of a line in R 3 would be.

Vectors and the Geometry of Space

Eqaution of Lines and Planes.  Determine the vector and parametric equations of a line that contains the point (2,1) and (-3,5).

10.5 Lines and Planes in Space Parametric Equations for a line in space Linear equation for plane in space Sketching planes given equations Finding distance.

Vectors in 2-Space and 3-Space II

Section 13.4 The Cross Product. Torque Torque is a measure of how much a force acting on an object causes that object to rotate –The object rotates around.

The Vector Product. Remember that the scalar product is a number, not a vector Using the vector product to multiply two vectors, will leave the answer.

Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to the plane. It can be found by the Cross product of any.

Section 3.5 Lines and Planes in 3-Space. Let n = (a, b, c) ≠ 0 be a vector normal (perpendicular) to the plane containing the point P 0 (x 0, y 0, z 0.

Mathematics. Session Three Dimensional Geometry–1(Straight Line)

Specialist Maths Vectors and Geometry Week 4. Lines in Space Vector Equation Parametric Equations Cartesian Equation.

1.3 Lines and Planes. To determine a line L, we need a point P(x 1,y 1,z 1 ) on L and a direction vector for the line L. The parametric equations of a.

Chapter 2 Section 2.4 Lines and Planes in Space. x y z.

Assigned work: pg. 443 #2,3ac,5,6-9,10ae,11,12 Recall: We have used a direction vector that was parallel to the line to find an equation. Now we will use.

Vector Space o Vector has a direction and a magnitude, not location o Vectors can be defined by their I, j, k components o Point (position vector) is a.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Chapter 12: Vectors Cartesian.

Equations of Lines and Planes

1. Given vectors a, b, and c: Graph: a – b + 2c and 3c – 2a + b 2. Prove that these following vectors a = 3i – 2j + k, b = i – 3j +5k, and c = 2i +j –

Graphing in 3-D Graphing in 3-D means that we need 3 coordinates to define a point (x,y,z) These are the coordinate planes, and they divide space into.

DOT PRODUCT CROSS PRODUCT APPLICATIONS

Find the equation of the line with: 1. m = 3, b = m = -2, b = 5 3. m = 2 (1, 4) 4. m = -3 (-2, 8) y = 3x – 2 y = -2x + 5 y = -3x + 2 y = 2x + 2.

Further vectors. Vector line equation in three dimensions.

Warm-Up 5 minutes 1. Graph the line y = 3x + 4.

X = 2 + t y = t t = x – 2 t = (y + 3)/2 x – 2 = y x – 4 = y + 3 y – 2x + 7 = 0 Finding the Cartesian Equation from a vector equation x = 2.

CSE 681 Brief Review: Vectors. CSE 681 Vectors Basics Normalizing a vector => unit vector Dot product Cross product Reflection vector Parametric form.

Parametric and general equation of a geometrical object O EQUATION INPUT t OUTPUT EQUATION INPUT OUTPUT (x, y, z)

3D Lines and Planes.

Extended Work on 3D Lines and Planes. Intersection of a Line and a Plane Find the point of intersection between the line and the plane Answer: (2, -3,

Yashavantrao Chavan Institute of Science Satara. Rayat Shikshan Sanstha’s Rayat Gurukul CET Project Std : XII Sub : -Mathematics.

LECTURE 5 OF 8 Topic 5: VECTORS 5.5 Application of Vectors In Geometry.

CSE 681 Brief Review: Vectors. CSE 681 Vectors Direction in space Normalizing a vector => unit vector Dot product Cross product Parametric form of a line.

Normal Vector. The vector Normal Vector Definition is a normal vector to the plane, that is to say, perpendicular to the plane. If P(x0, y0, z0) is a.

Lines and Planes In three dimensions, we use vectors to indicate the direction of a line. as a direction vector would indicate that Δx = 7, Δy = 6, and.

Vectors in space Two vectors in space will either be parallel, intersecting or skew. Parallel lines Lines in space can be parallel with an angle between.

Warm up 1.) (3, 2, -4), (-1, 0, -7) Find the vector in standard position and find the magnitude of the vector.

Chapter 2 Planes and Lines

2 Vectors In 2-space And 3-space

Lesson 12 – 10 Vectors in Space

LESSON 90 – DISTANCES IN 3 SPACE

Lesson 3-6: Perpendicular & Distance

Planes in Space.

11 Vectors and the Geometry of Space

Intersection between - Lines, - Planes and - a plane & a Line

Parallel and Perpendicular Lines

Lines and Planes in Space

Math 200 Week 3 - Monday Planes.

Parallel and Perpendicular Lines

C H A P T E R 3 Vectors in 2-Space and 3-Space

Notation of lines in space

Find a vector equation for the line through the points {image} and {image} {image}

Find a vector equation for the line through the points {image} and {image} {image}

By the end of Week 3: You would learn how to solve many problems involving lines/planes and manipulate with different coordinate systems. These are.

Lesson 83 Equations of Planes.

Chapter 3-6 Perpendiculars and Distance

Topic Past Papers – Vectors

2 Vectors in 2-space and 3-space

11 Vectors and the Geometry of Space

Vector Equations in Space

CHAPTER 3 VECTORS NHAA/IMK/UNIMAP.

Presentation transcript:

VECTORS APPLICATIONS NHAA/IMK/UNIMAP

APPLICATIONS Lines & Line Segment in Space Parametric Equations z L v y x P0(x0,y0,z0) L P(x,y,z) v Line L is the set of all points P(x,y,z) for which parallel to NHAA/IMK/UNIMAP

APPLICATIONS Therefore the parametric equation for L : Cartesian equation: Parametric Equation NHAA/IMK/UNIMAP

Example 1 Find parametric and Cartesian equation for the line passes through Q(-2,0,4) and parallel to NHAA/IMK/UNIMAP

Example 2 Find the parametric equation for the line passes through P(-3,2,-3) and Q(1,-1,4) NHAA/IMK/UNIMAP

APPLICATIONS Distance from point S to line L L v L From the properties of Cross Product Formula of Distance from point S to L NHAA/IMK/UNIMAP

Example 3 Find the distance from the point S(1,1,5) to the line NHAA/IMK/UNIMAP

APPLICATIONS Equation of Planes P0(x0,y0,z0) P(x,y,z) Vector is on the plane M and vector which is perpendicular to M known as normal vector, n Equation of Plane is defined by: From the properties of Dot Product NHAA/IMK/UNIMAP

APPLICATIONS Equation of Planes Normal vector n : NHAA/IMK/UNIMAP

APPLICATIONS Let and EQUATION OF PLANE NHAA/IMK/UNIMAP

Example 4 Find an equation of plane through P0(-3,0,7) perpendicular to NHAA/IMK/UNIMAP

Example 7: Find equation of plane through 3 points: NHAA/IMK/UNIMAP

APPLICATIONS Lines of Intersection 2 planes parallel if and only if their normal planes are parallel 2 planes that are not parallel, intersect in a line nM nN Normal vector for plane M and N v Line of intersection NHAA/IMK/UNIMAP

APPLICATIONS Lines of Intersection nM Finding v : nN v Line of intersection NHAA/IMK/UNIMAP

Example 8: Find a vector parallel to the line of intersection of the planes NHAA/IMK/UNIMAP

Example 9: Find the parametric equation for the line in which the planes intersect. NHAA/IMK/UNIMAP

APPLICATIONS Distance from a Point to the Plane P n D P0 NHAA/IMK/UNIMAP

APPLICATIONS Equation of plane NHAA/IMK/UNIMAP

Example 10: Find the distance from S(1,1,3) to the plane NHAA/IMK/UNIMAP