Tài liệu Chapter 3: Vectors docx

Chapter Vectors In Physics we have parameters that can be completely described by a number and are known as “scalars” Temperature, and mass are such parameters Other physical parameters require additional information about direction and are known as “vectors” Examples of vectors are displacement, velocity and acceleration In this chapter we learn the basic mathematical language to describe vectors In particular we will learn the following: Geometric vector addition and subtraction Resolving a vector into its components The notion of a unit vector Add and subtract vectors by components Multiplication of a vector by a scalar The scalar (dot) product of two vectors The vector (cross) product of two vectors (3-1) An example of a vector is the displacement vector which describes the change in position of an object as it moves from point A to point B This is represented by an arrow that points from point A to point B The length of the arrow is proportional to the displacement magnitude The direction of the arrow indicated the displacement direction The three arrows from A to B, from A' to B', and from A'' to B'', have the same magnitude and direction A vector can be shifted without changing its value if its length and direction are not changed In books vectors are written in two ways:  Methoda1: Method 2: (3-2) a (using an arrow above) (using bold face print) The magnitude of the vector is indicated by italic print: a Geometric vector Addition    s a  b  Sketch vector a using an appropriate scale  Sketch vector b using the same scale   Place the tail of b at the tip of a   The vector s starts from the tail of a  and terminates at the tip of b Vector addition is commutative     a  b b  a   Negative  b of a given vector b    b has the same magnitude as b but opposite direction (3-3) Geometric vector Subtraction    d a  b      We write: d a  b a   b   From vector b we find  b   Then we add  b to vector a     We thus reduce vector subtraction to vector addition which we know how to Note: We can add and subtract vectors using the method of components For many applications this is a more convenient method (3-4) A component of a vector along an axis is the projection of the vector on this axis For example ax is the  projection of a along the x-axis The component ax C A B is defined by drawing straight lines from the tail  and tip of the vector a which are perpendicular to the x-axis From triangle ABC the x- and y-components  of vector a are given by the equations: ax a cos  , a y a sin  If we know ax and a y we can determine a and  From triangle ABC we have: ay 2 a  ax  a y , tan   ax (3-5) Unit Vectors (3-6) A unit vector is defined as vector that has magnitude equal to and points in a particular direction A unit vector is defined as vector that has magnitude equal to and points in a particular direction Unit vector lack units and their sole purpose is to point in a particular direction The unit vectors along the x, y, and z axes are labeled iˆ , ˆj , and kˆ, respectively Unit vectors are used to express other vectors  For example vector a can be written as:  a ax iˆ  a y ˆj The quantities axiˆ and a y ˆj are called  the vector components of vector a y  r  a  b Adding Vectors by Components x O   We are given two vectors a ax iˆ  a y ˆj and b bxiˆ  by ˆj  We want to calculate the vector sum r rx iˆ  ry ˆj The components rx and ry are given by the equations: rx ax  bx and ry a y  by (3-7) y  d  b  a Subtracting Vectors by Components x O   ˆ ˆ We are given two vectors a ax i  a y j and b bxiˆ  by ˆj We want to calculate the vector difference    d  a  b d xiˆ  d y ˆj  The components d x and d y of d are given by the equations: d x ax  bx and d y a y  by (3-8) Multiplying a Vector by a Scalar    Multiplication of a vector a by a scalar s r esults in a new vector b sa The magnitude b of the new vector is given by: b  | s | a   If s  vector b has the same direction as vector a   If s  vector b has a direction opposite to that of vector a The Scalar Product of two Vectors     The scalar product a b of two vectors a and b is given by:   a b =ab cos  The scalar product of two vectors is also known as the "dot" product The scalar product in terms of vector components is given by the equation:   a b =axbx  a y by  az bz (3-9) The Vector Product of two Vectors      The vector product c a b of the vectors a and b  is a vector c  The magnitude of c is given by the equation: c ab sin   The direction of c is perpendicular to the plane P defined   by the vectors a and b  The sense of the vector c is given by the right hand rule:   a Place the vectors a and b tail to tail  b Rotate a in the plane P along the shortest angle  so that it coincides with b c Rotate the fingers of the right hand in the same direction  d The thumb of the right hand gives the sense of c The vector product of two vectors is also known as (3-10) the "cross" product    The Vector Product c a b in terms of Vector Components    ˆ ˆ ˆ ˆ ˆ ˆ a a x i  a y j  az k , b b x i  by j  bz k , c c x iˆ  c y ˆj  cz kˆ  The vector components of vector c are given by the equations: cx a y bz  az by , c y az bx  ax bz , c z axby  a y bx Note: Those familiar with the use of determinants can use the expression i j k   a b  ax a y az bx by bz Note: The order of the two vectors in the cross product is important     b a  a b (3-11)   ... point in a particular direction The unit vectors along the x, y, and z axes are labeled iˆ , ˆj , and kˆ, respectively Unit vectors are used to express other vectors  For example vector a can be... vector a The Scalar Product of two Vectors     The scalar product a b of two vectors a and b is given by:   a b =ab cos  The scalar product of two vectors is also known as the "dot"... equations: rx ax  bx and ry a y  by (3-7) y  d  b  a Subtracting Vectors by Components x O   ˆ ˆ We are given two vectors a ax i  a y j and b bxiˆ  by ˆj We want to calculate the vector