Chapter 3: Vectors

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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.

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Nội dung Text: Chapter 3: Vectors

Chapter 3 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 (3-1) The vector (cross) product of two vectors
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: r Methoda 1: (using an arrow above) Method 2: a (using bold face print) (3-2) The magnitude of the vector is indicated by italic print: a
Geometric vector Addition r r r s = a +b r Sketch vector a using an appropriate scale r Sketch vector b using the same scale r r Place the tail of b at the tip of a r r The vector s starts from the tail of a r and terminates at the tip of b Vector addition is commutative r r r r a +b = b +a r r Negative − b of a given vector b r r −b has the same magnitude as b but opposite direction (3-3)
Geometric vector Subtraction r r r d = a −b r r r r r ( ) We write: d = a − b = a + −b r r From vector b we find − b r ( ) Then we add −b to vector a r We thus reduce vector subtraction to vector addition which we know how to do 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 r projection of a along the x-axis. The component ax is defined by drawing straight lines from the tail r and tip of the vector a which are perpendicular to the x-axis. C From triangle ABC the x- and y-components r of vector a are given by the equations: A B ax = a cos θ , a y = a sin θ If we know ax and a y we can determine a and θ . From triangle ABC we have: ay a = ax + a y 2 2 , tan θ = ax (3-5)
Unit Vectors (3-6) A unit vector is defined as vector that has magnitude equal to 1 and points in a particular direction. A unit vector is defined as vector that has magnitude equal to 1 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 ˆ are labeled i , ˆ, and k , respectively. ˆ j Unit vectors are used to express other vectors r For example vector a can be written as: r a = ax i + a y ˆ . ˆ j The quantities ax i and a y ˆ are called ˆ j r the vector components of vector a
y r r r Adding Vectors by Components b r a x O r r We are given two vectors a = ax i + a y ˆ and b = bx i + by ˆ ˆ j ˆ j r We want to calculate the vector sum r = rx i + ry ˆ ˆ j The components rx and ry are given by the equations: rx = ax + bx and ry = a y + by (3-7)
y r r d Subtracting Vectors by Components b r a x O r r ˆ + a y ˆ and b = bx i + by ˆ We are given two vectors a = ax i j ˆ j We want to calculate the vector difference r r r d = a − b = d xi + d y ˆ ˆ j r 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 r r r 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 r r If s > 0 vector b has the same direction as vector a r r If s < 0 vector b has a direction opposite to that of vector a The Scalar Product of two Vectors r r r r The scalar product a ⋅ b of two vectors a and b is given by: r r a ⋅ b =ab cos φ 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: r r a ⋅ b =axbx + a y by + az bz (3-9)
The Vector Product of two Vectors r r r r r The vector product c = a × b of the vectors a and b r is a vector c r The magnitude of c is given by the equation: c = ab sin φ r The direction of c is perpendicular to the plane P defined r r by the vectors a and b r The sense of the vector c is given by the right hand rule: r r a. Place the vectors a and b tail to tail r b. Rotate a in the plane P along the shortest angle r so that it coincides with b c. Rotate the fingers of the right hand in the same direction r 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
r r r The Vector Product c = a × b in terms of Vector Components r r a = a xi j ˆ j ˆ r ˆ j ˆ ˆ + a y ˆ + az k , b = b x i + by ˆ + bz k , c = c x i + c y ˆ + cz k ˆ r 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 = ax by − a y bx Note: Those familiar with the use of determinants can use the expression $ i $j k$ r r a × b = ax a y az bx by bz Note: The order of the two vectors in the cross product is important r r r r ( b ×a = − a ×b ) (3-11)