Vector scalar: a physical scalar with only size and no direction: we call it scalar. Quantity, we call it scalar. Vector: there is a physical quantity, vector: there is a physical quantity, which can not be completely described only by size, but also by direction. For example, it needs to be described by direction. For example, we only know that a person walked 1 km from the school gate, and we can't be sure where he went. But if you also know that the direction he is going is due east, that place. But if we know that he is going due east, we can determine where he has arrived. This has both size and direction, so as to judge where he has arrived. This physical quantity with both magnitude and direction is called a vector. We call it a vector. The fundamental difference between a vector and a scalar is whether there is a direction. The fundamental difference between a vector and a scalar is whether there is a direction. Modulus of a vector: The size of a vector is called the modulus of a vector. Modulus of Vector A: The size of the vector is called the modulus of the vector. The modulus of v is:, and the modulus of v is: a or | A |. Vector has translation invariance: vector has translation invariance: the size and direction of vector will not change when it is translated in space, and the size and direction of vector will not change when it is translated in space. This property is called vector translation invariance. It is called vector translation invariance. The representation of two vectors in cartesian coordinates: the representation of one vector: the representation of V vector A in cartesian coordinates can be represented by its three projection components (AX, Ay, Az) in cartesian coordinates: v v v unit vectors point to the positive directions of the three coordinate axes respectively. I, j, k: unit vectors, pointing to the positive directions of the three coordinate axes respectively. The expression of v v v A = Ax i+A y j+Az k in spherical coordinates: v v A = AeA v v v where: is the module of vector A, and eA is the unit vector ... vector pointing in the direction of vector A. Cosine in V direction: The positive angles α, β and γ formed by a vector A and three coordinate axes cosine: V in rectangular coordinates are called the direction cosine of vector A ... Obviously: cosine. It is obvious that: AYAX AZ COS α = COS γ = COS β = AAAA V V V In the Physics Lecture of Qingdao University of Science and Technology 1, the direction coda A = A(cos α i+cos β j+cos γ k) string is used to represent the synthesis of three vectors. Vector addition. V V V V V V A+B =(AXI+AYJ+AZK)+(BX I+BYJ+BZK)V V V =(AX+BX)I+(AY+BY)J+(AZ+BZ)K 2。 Vector subtraction Vector subtraction (subtraction) v v V is opposite in direction and equal in size. B and B are opposite in direction and equal in size, including: v v v v v? B =？ Bx i？ B y j？ Bz k Qingdao university of science and technology university physics lecture notes vector subtraction V V V V V V V A? B = ( Ax i + Ay j + Az k)？ (Bx i + B y j + Bz k ) v v v = ( Ax？ Bx )i + ( Ay？ B y ) j + ( Az？ Bz )k the addition and subtraction of k vector is called the addition and subtraction of vector composition vector is called vector composition. The scalar product of a vector is also called the point multiplication of a vector, and the scalar product of a vector is also called the point multiplication of a vector. Defined as sum) In essence, the product of the size of one vector and the projection size of another vector in its direction, v v A B = AB cos α v v v v v v V V is defined as: i i = j j = k k = 1, And the scalar product is defined as: V V V V V V V V V V I J = J I = K.I = J K = 0 The scalar product of the vector of college physics lectures obeys (1) exchange rate: (2) combined exchange rate: V V V V AB = B V V V V V V V V V V (A+B) C. Vector product vector product. The cross product of a vector is also called the cross product of a vector, and it is defined as: the cross product of a vector is also called the cross product of a vector, and it is defined as: the unit vector determined by the cross product V according to the right-hand spiral law. Where e is the unit vector determined by a and b according to the right-hand helix law. Definition of slave vector product: definition of slave vector product: VVA× B = AB sin α eV V V V V V V V V V I× J = K× K = 0 University Physics Lecture Qingdao University of Science and Technology v v i× j = k v v v j× i =? K memory mode v v v j x k = I v v k x j =? i v v v k ×i = j v v v i ×k =？ j v v v v v v v v v v v I？ j？ k？ Me? j？ k？ Me? j？ k？ The forward cross product is positive and the reverse cross product is negative. The forward cross product is positive and the reverse cross product is negative. The cross product has the following properties: (1) Disobeying the exchange rate: Disobeying the exchange rate v v v a× b =? B × A Pay attention to the right-handed spiral rotation law of the coordinate axis. V v v v (2) Observation distribution rate: C × (A+B) = C × A+C × B Observation distribution rate: (3) Vector product of two parallel or antiparallel vectors is 0. The vector product of two parallel or antiparallel vectors is. Lecture Notes on College Physics of Qingdao University of Science and Technology 5 1. The differential vector of vector calculus vector can only be applied to the derivative formula of scalar: it can only be applied to the derivative formula of scalar: v v d v v dA dB (1) (A+B). =+dt dt dt v d[f(t)A]df(t)v dA(2)= A+f(t)dt dt dt v v v dB dA v d v v B(3)(A B)= A+dt dt dt dt v v v dB dA v d v v v(4)(A×B)= A×+×B dt。 As a special case of formula (1), vector in cartesian coordinates: as a special case of formula in cartesian coordinates: V V V A = AXI+AYJ+AZK as a special case of formula (2), vector in spherical coordinates: as an example of formula in spherical coordinates:. Integral (integral) of integral vector vs.: (1) time integral t:) ∫ t2t1vt2vvadt = ∫ (axi+ayj+azk) DTT1= (∫ t2t/kloc) The line integral of: (2) along the curve S:) ∫ S V VS = ∫ AXDX+∫ Aydy+∫ AZDZ X1Y1KLOC-0/Qingdao University of Science and Technology's six-position vector reference system. 1 reference system The standard object used to describe the motion of an object is called the reference system. Different reference frames are selected, and the descriptions of object motion are different, and the descriptions of object motion are also different. This is the relativity of motion description. This is the relativity of motion description. Example 2 (material point, particle) particle (when studying the motion of an object, if its size and shape can be ignored, if its influence on the motion of the object can be ignored, the object can be regarded as a point (particle) with mass. A point (particle) with mass is an idealized physical model formed by scientific abstraction. The purpose is to highlight the main nature of the research object. Let's not consider some minor factors for the time being. The physical quantity that determines the position of the particle P at a certain moment in the coordinate system is called V. In the straight position vector, the position vector R is in rectangular coordinates for short, and its expression is: in angular coordinates, its expression is: y y v j v * p r v v v v r = xi+y j+ZK v v v j k, where i and are unit vectors on the X, Y and Z axes respectively. The size of the unit vector in the axial direction (the modulus is the modulus of the potential vector r). For the physics lecture notes of Qingdao University of Science and Technology v 2 2 2 r = r = x+y +z, the trajectory equation of direction cosine cosα = x r cos β = y r cos γ = z r y β v r α P P o 4 (trajectory equation). The equation of the trajectory equation curve can be expressed as z γ x v r (t) y y (t) f (x, y, Z) = 0, and it can also be expressed as the x(t) x component formula (parameter equation parameter equation) parameter equation x = x(t) y = y (t) z = z (t in the rectangular coordinates. I+y(t) j+z(t)k From the vector form of trajectory equation, from the vector form of trajectory equation, we can immediately write the component formula of curve equation, and we can get the familiar curve equation f (x, y, Z) = 0. For example, if the parameter equation of spiral is known as x = a cosθ, its vector equation is y = a sin θ z = b θ. The function of a sin θ i+a cosθ j+bk dθ, if the independent variable θ is still a function of time t, then there are v v v r = a cos θ i+a sin θ j+b θ k v dr d θ v d θ v d θ v d θ v d θ v =? A sin θ? i + a cosθ？ j +b？ K dt dt the theory of relativity described in the physics handout of Qingdao University of Science and Technology, the mother got on a bus with her obedient son and the mother got on a bus with her obedient son. When the bus was driving at a high speed, the mother said to her son, "Stand still." The son obeyed, and the mother said to her son, stand still. Stand well, then if the mother is the frame of reference, the son is still, stand well, then if the mother is the frame of reference, the son is still, and if the ground observer is the frame of reference, the child is moving. If the observer on the ground is taken as the frame of reference, the child is moving.