Vectors in Space 11.2 JMerrill, 2010. Rules The same rules apply in 3-D space: The component form is found by subtracting the coordinates of the initial.

Slides:

Advertisements

Similar presentations

Vectors Maggie Ambrose Maddy Farber. Hook… Component Form of a Vector  If v is a vector in a plane whose initial point is the origin and whose terminal.

Advertisements

Chapter 10 Vocabulary.

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

11 Analytic Geometry in Three Dimensions

The Dot Product (MAT 170) Sections 6.7

10.6 Vectors in Space.

Digital Lesson Vectors.

10.5 The Dot Product. Theorem Properties of Dot Product If u, v, and w are vectors, then Commutative Property Distributive Property.

Vectors Sections 6.6.

Vectors Sections 6.6. Objectives Rewrite a vector in rectangular coordinates (in terms of i and j) given the initial and terminal points of the vector.

The Cross Product of 2 Vectors 11.3 JMerrill, 2010.

6.4 Vectors and Dot Products

8.6.1 – The Dot Product (Inner Product). So far, we have covered basic operations of vectors – Addition/Subtraction – Multiplication of scalars – Writing.

Vectors in the Plane Peter Paliwoda.

Multiplication with Vectors

6.4 Vectors and Dot Products The Definition of the Dot Product of Two Vectors The dot product of u = and v = is Ex.’s Find each dot product.

1.1 – 1.2 The Geometry and Algebra of Vectors.  Quantities that have magnitude but not direction are called scalars. Ex: Area, volume, temperature, time,

VECTOR CALCULUS. Vector Multiplication b sin   A = a  b Area of the parallelogram formed by a and b.

Vectors in the Plane Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A ball flies through the air at a certain speed.

H.Melikyan/12001 Vectors Dr.Hayk Melikyan Departmen of Mathematics and CS

Section 10.2a VECTORS IN THE PLANE. Vectors in the Plane Some quantities only have magnitude, and are called scalars … Examples? Some quantities have.

Vectors A quantity which has both magnitude and direction is called a vector. Vector notations A B a AB = a AB x and y are the components of vector AB.

Vectors Vectors are represented by a directed line segment its length representing the magnitude and an arrow indicating the direction A B or u u This.

Lesson  Find scalar products of two-space vectors  Prove or disprove generalizations about vector operations  Identify parallel and orthogonal.

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

Vectors in Plane Objectives of this Section Graph Vectors Find a Position Vector Add and Subtract Vectors Find a Scalar Product and Magnitude of a Vector.

HWQ Find the xy trace:.

Copyright © Cengage Learning. All rights reserved. 10 Topics in Analytic Geometry.

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.

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

Warm UpMar. 14 th 1.Find the magnitude and direction of a vector with initial point (-5, 7) and terminal point (-1, -3) 2.Find, in simplest form, the unit.

8.1 and 8.2 answers. 8.3: Vectors February 9, 2009.

Meeting 23 Vectors. Vectors in 2-Space, 3-Space, and n- Space We will denote vectors in boldface type such as a, b, v, w, and x, and we will denote scalars.

11.2 Space coordinates and vectors in Space. 3 dimensional coordinate plane.

Adding, Subtracting and Parallel Vectors Lessons 10.5 – 10.6.

12.2 Vectors.  Quantities that have magnitude but not direction are called scalars. Ex: Area, volume, temperature, time, etc.  Quantities such as force,

Copyright © Cengage Learning. All rights reserved. 12 Vectors and the Geometry of Space.

Vectors and Dot Product 6.4 JMerrill, Quick Review of Vectors: Definitions Vectors are quantities that are described by direction and magnitude.

A.) Scalar - A single number which is used to represent a quantity indicating magnitude or size. B.) Vector - A representation of certain quantities which.

Section 4.2 – The Dot Product. The Dot Product (inner product) where is the angle between the two vectors we refer to the vectors as ORTHOGONAL.

The Dot Product. Note v and w are parallel if there exists a number, n such that v = nw v and w are orthogonal if the angle between them is 90 o.

6.4 Vector and Dot Products. Dot Product  This vector product results in a scalar  Example 1: Find the dot product.

12.3 The Dot Product. The dot product of u and v in the plane is The dot product of u and v in space is Two vectors u and v are orthogonal  if they meet.

Section 6.3. A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The.

6.4 Vectors and Dot Products Objectives: Students will find the dot product of two vectors and use properties of the dot product. Students will find angles.

11. Section 12.1 Vectors Vectors What is a vector and how do you combine them?

Vectors Def. A vector is a quantity that has both magnitude and direction. v is displacement vector from A to B A is the initial point, B is the terminal.

Dot Product and Cross Product. Dot Product  Definition:  If a = and b =, then the dot product of a and b is number a · b given by a · b = a 1 b 1 +

CHAPTER 3 VECTORS NHAA/IMK/UNIMAP.

Analytic Geometry in Three Dimensions

6.2 - Dot Product of Vectors

Outline Addition and subtraction of vectors Vector decomposition

Vectors in the Plane.

Scalars and Vectors.

8.5 The Dot Product.

6.3-Vectors in the Plane.

3D Coordinates In the real world points in space can be located using a 3D coordinate system. For example, air traffic controllers find the location a.

Section 3.2 – The Dot Product

12.3 The Dot Product.

Scalars Some quantities, like temperature, distance, height, area, and volume, can be represented by a ________________ that indicates __________________,

Only some of this is review.

Vectors in space - 3 dimensions

Ch. 15- Vectors in 2-D.

2 Vectors in 2-space and 3-space

Vectors and Dot Products

Digital Lesson Vectors in the Plane.

CHAPTER 3 VECTORS NHAA/IMK/UNIMAP.

Math with Vectors I. Math Operations with Vectors.

Presentation transcript:

Vectors in Space 11.2 JMerrill, 2010

Rules The same rules apply in 3-D space: The component form is found by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point. Two vectors are = iff their corresponding components are =. The magnitude (length) of v Vector addition still means you add the value in the x place, the y place, and the z place

Scalar multiplication still means you distribute the scalar over the component form The dot product of If the dot product = 0, then the vectors are orthogonal. The angle between two nonzero vectors u and v is

Finding the Component Form of a Vector Find the component form and magnitude of vector v having an initial point of (3,4,2) and a terminal point of (3,6,4) The component form is The magnitude is

Finding the Dot Product of Two Vectors Find the dot product of and Remember the dot product is a scalar, not a vector.

Find the Angle Between Two Vectors Find the angle between u = and v =

Parallel Vectors Vector w has an initial point (1,-2,0) and a terminal point (3,2,1). Which of the following vectors is parallel to w? A. u = <4,8,2> B. v = <4,8,4> Put w into component form <2,4,1> Vector u is the answer because it is just double vector w. U can be written as 2<2,4,1>

Using Vectors to Determine Collinear Points Determine whether the points P(2,-1,4), Q(5,4,6), and R(-4,-11,0) are collinear. The points P,Q, and R are collinear iff the vectors PQ and PR are parallel. Because PR = -2PQ, the vectors are parallel. Therefore, the points are collinear.

Finding the Terminal Point of a Vector The initial point of v = is P(3,-1,6). What is the terminal point? We use the initial point of P(3,-1,6) and the terminal point of Q(q 1, q 2, q 3 ) PQ = = The 1 st term = the 1 st term, the 2 nd term = the 2 nd term… So, q 1 –3 = 4, q 2 -(-1) = 2, q 3 –6 = -1 Point Q = (7, 1, 5)