2 Vectors In 2-space And 3-space

Overview In this chapter we review the related concepts of physical vectors, geometric vectors, and algebraic vectors. To provide maximum geometric insight, we concentrate on vectors in two-space and three-space. Later, in Chapter 3, we will generalize many of the ideas developed in this chapter and apply them to a study of vectors in n-space, that is, to vectors in Rn. A major emphasis in Chapter 3 is on certain fundamental ideas such as subspaces of Rn and the dimension of a subspace. As we will see in Chapter 3, concepts such as subspace and dimension are directly related to the geometrically familiar notions of lines and planes in three-space.

Core sections Vectors in the plane Vectors in space The dot product and the cross product Lines and planes in space

2.1 Vectors in the plane 1. Three types of vectors Physical vectors: A physical quantity having both magnitude and direction is called a vector. Typical physical vectors are forces, displacements, velocities, accelerations.

(2) Geometric vectors: The directed line segment from point A to point B is called a geometric vector and is denoted by For a given geometric vector ,the endpoint A is called the initial point and B is the terminal point.

(3) Equality of geometric vectors All geometric vectors having the same direction and magni-tude will be regarded as equal, regardless of whether or not they have the same endpoints. x y E F A B C D

(4) Position vectors x y A B O P

(5) Components of a vector

(6) An equality test for Geometric Vectors Theorem2.1.1: Let and be geometric vectors. Then if and only if their components are equal.

(7) Algebraic vectors: Theorem2.1.2: Let be a geometric vector, with A=(a1,a2) and B=(b1,b2). Then can be represented by the algebraic vector

2. Using algebraic vectors to calculate the sum of geometric vectors Theorem2.1.2: Let u and v be geometric vectors with algebraic representations given by Then the sum u+v has the following algebraic representation:

3. Scalar multiplication Theorem2.1.3: Let u be a geometric vectors with algebraic representations given by Then the scalar multiple cu has the following algebraic representation:

2.1 Exercise P126 26 4. Subtracting geometric vectors 5. Parallel vectors Vectors u and v are parallel if there is a nonzero scalar c such that v=cu. If c>0, we say u and v have the same direction but if c<0, we say u and v have the opposite direction. 6. Lengths of vectors and unit vectors 7. The basic vectors i and j 2.1 Exercise P126 26

2.2 Vectors in space 1. Coordinate axes in three space 2. The right-hand rule 3. Rectangular coordinates for points in three space axis;coordinate planes;octants 4. The distance formula Theorem2.2.1: Let P=(x1,y1,z1) and Q=(x2,y2,z2) be two points in three space. The distance between P and Q, denoted by d(P,Q), is given by

6. Geometric vectors and their components 5. The midpoint formula Theorem2.2.2: Let P=(x1,y1,z1) and Q=(x2,y2,z2) be two points in three space. Let M denote the midpoint of the line segment joining P and Q. Then, M is given by 6. Geometric vectors and their components 7. Addition and scalar multiplication for vectors 8. Parallel vectors, lengths of vectors, and unit vectors 9. The basic unit vectors in three space

2.3 The dot product and the cross product 1. The dot product of two vectors Definition 2.3.1: Let u and v are vectors, then the dot product of u and v, denoted u·v, is defined by u · v=||u|| ||v|| cosθ. where θis the angle of vectors u and v. Definition 2.3.2: Let u and v are two-dimensional vectors, then the dot product of u and v, denoted u·v, is defined by u · v=u1v1+u2v2. Let u and v are three-dimensional vectors, then the dot product of u and v, denoted u·v, is defined by u · v=u1v1+u2v2+ u3v3.

2. The angle between two vectors u · v=||u|| ||v|| cosθ. 3. Algebraic properties of the dot product

4. Orthogonal Vectors(正交向量） When θ=π/2 we say that u and v are perpendicular or orthogonal. Theorem 2.3.1: Let u and v are vectors, then u and v are orthogonal if and only if u · v=0. In the plane, the basic unit vectors i and j are orthogonal. In three space, the basic unit vectors i, j and k are mutually orthogonal.

||u×v||=||u|| ||v|| sinθ. q v u θ 5. Projections 6. The cross product Definition 2.3.3: Let u and v are vectors, then the cross product of u and v, denoted u×v, is a vector that it is orthogonal to u and v, and u,v,u×v is right-hand system, and the norm of the vector is ||u×v||=||u|| ||v|| sinθ. where θis the angle of vectors u and v.

7. Remember the form of the cross product (two methods)

8. Algebraic properties of the cross product 9. Geometric properties of the cross product

2.3 Exercise P148 48 10. Triple products（三重积） 11. Tests for collinearity and coplanarity Theorem: Let u, v and w be nonzero three dimensional vectors. u and v are collinear if and only if u×v=0. u, v and w are coplanar if and only if u·(v×w). 2.3 Exercise P148 48

2.4 Lines and planes in space 1. The equation of a line in xy-plane y x O P0=(x0,y0) . l

2. The equation of a line in three space (1)Point and directional vector form equation of a line

(2)Parametric equations of a line

Example1: Let L be the through P0=(2,1,6), having direction vector u given by u=[4,-1,3]T. Find parametric equations for the line L. Does the line L intersect the xy-plane? If so, what are the coordinates of the point of intersection? Example2: Find parametric equations for the line L passing through P0=(2,5,7) and the point P1=(4,9,8).

2. The equation of a plane in three space Point and normal vector form equation of a plane

Example3: Find the equation of the plane containing the point P0=(1,3,-2) and having normal n=[5,-2,2]T. Example4: Find the equation of the plane passing through the points P0=(1,3,2), P1=(2,0,-1), and P2=(4,5,1).

The relationship between two lines or two planes A line and a plane Two plane