Lecture notes of Physics The Vectors Polaris Coordinates of a Vector How can we represent a vector? -We plot an arrow: • the length proportional to magnitude of vector • the line represents the vector direction • the point represents the vector path e.g. we plot a vector with: inclination p/6 direction: upwards path: upwards magnitude: 55 uu Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected] 5 4 3 2 1 p/6 The Polaris Coordinates are (magnitude, inclination) (r, a) e.g. (5, p/6) Lecture notes of Physics The Vectors Cartesian Coordinate of a Vector We can use the Cartesians Axes for represent a vector e.g. we plot the previous vector: direction: slope p/6 path: upwards magnitude: 5 u y (r·cos(a), r·sin(a)) r·sin(a) a=p/6 O We can write the vector as v = ρ cosa ˆi ρ sina ˆj x r·cos(a) The Cartesians Coordinates are (x-coordinate, y-coordinate) (x, y) e.g. (r·cos(a), r·sin(a)) r·cos(a) and r·sin(a) are known as Cartesian Components Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected] Change of Coordinate: Cartesians Polaris Lecture notes of Physics The Vectors y (r·cos(a), r·sin(a)) r·sin(a) a=p/6 O 5 (x, y) 4 x 3 2 r·cos(a) Cartesians Coordinates 1 p/6 (r, a) Polaris Coordinates ρ = x2 y2 From Cartesian to Polar x = r cosa y = r sin a y a = arctg x From Polar to Cartesian Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected] Lecture notes of Physics The Vectors Calculation with Vectors • Product with a scalar (k) a = k b b k=3 a a has the same direction and path as b and magnitude of a is three times greater than the magnitude of b if 0<k<1 the magnitude of a is smaller than the magnitude of b if k<0 the path of a is the opposite of path of b b Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected] k = -1 a Lecture notes of Physics The Vectors Calculation with Vectors • Addition of vectors s = a b b The Parallelogram Law a s + We calculate the magnitude of s with Carnot Theorem a s s2 = a2 + b2 – 2ab cos(a) N.B. s Liceo Scientifico Europeo ‘Rainerum’ is known as Gualtiero Giovanazzi [email protected] We always have s a+ b the Resultant of a and b Lecture notes of Physics The Vectors Calculation with Vectors y • Addition of vectors, Cartesian Coordinates s = a b b a + s·cos(a) a·cos(a) s·sin(a) O a·sin(a) b·cos(b) x s b·sin(b) We describe the vector sum s by adding the components of a and b s = xa xb iˆ ya yb ˆj Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected] Lecture notes of Physics The Vectors Calculation with Vectors • difference between vectors d = a b d =b a b a a d a - b b a - In both cases the magnitude of d is Liceo ‘Rainerum’ d2 =Scientifico a2 + b2 Europeo – 2ab cos(a) Gualtiero Giovanazzi [email protected] b d = ab a b d = ba a Proof We describe vector difference d by subtracting the components of b of a d = a b = xa xb iˆ ya yb ˆj Lecture notes of Physics The Vectors Calculation with Vectors Comparison between addition and difference of vectors • Addition s = ab s = a b • Difference d = a b magnitude of s is a s2 = a2 + b2 – Liceo Scientifico Europeo N.B.‘Rainerum’ Observe Gualtiero Giovanazzi [email protected] d = a b a magnitude of d is 2ab cos(a) s b b a a d2 = a2 + b2 – 2ab cos(a) d the position of a angle Lecture notes of Physics The Vectors Decomposition of a vector To decompose a vector v find two vectors v1, v2 in two prefixed direction, whose sum is equal to vector v v a2 v2 O v v1 b2 v1 a1 v c2 O v2 b1 v2 v d2 v2 O v1 c1 O d1 now from v we can draw two lines parallel to the axes in every case we can write: v = v1iˆ v2 ˆj v1 = v cos(angle between v and 1st axis) st Liceo Scientifico Europeo ‘Rainerum’ v2 = [email protected] v sin(angle between v and 1 axis) Gualtiero Giovanazzi Lecture notes of Physics The Vectors 1st Decomposition of a vector: Examples A body is hung to the ceiling with two different ropes. It forms angles a1, and a2 with the ropes. What are the tensions of the ropes? z2 T2 z2 T1 v2 v1 z1 z2 v The vector of body weight is directed downwards The vectors of tensions are direct toward the ropes So we can recognize two particular direction: the lines along the ropes So we draw a system of reference with the AXES parallel to the ropes and‘Rainerum’ ORIGINATING from Liceo Scientifico Europeo the body Gualtiero Giovanazzi [email protected] z1 z1 z2 z 1 v1 O v2 We can decompose v in directions z1, z2 to find 2 vectors, v1 and v2, whose sum is equal to v: v = v1 + v2 We can observe that T 1= -v1 and T2= -v2 Therefore v is balanced by the two vectors T1, T2 tensions of ropes: -v = T1 + T2 Lecture notes of Physics The Vectors Decomposition of a vector: Examples 2nd A body is sliding on an inclined plane. z2 v2 v2 z2 z2 v 1 v1 v z1 a z1 v2 What force pulls down the body? z2 Vector v directed downwards is the Force (or acceleration) of gravity v1 z1 z1 Vector v1 directed parallel to the plane is the active component of the Force (or acceleration) of gravity responsible of sliding of the body. It is indicated by v// v// = v · sin(a) The vector of body weight is directed downwards So we can recognize two particular direction: the line along the plane and the line they perpendicular to it Therefore we draw a system of reference with the AXES parallel and perpendicular to the plane We canScientifico decompose v in directions z1, z2 to find 2 Liceo Europeo ‘Rainerum’ vectors, v1 and v2, whose sum is equal to v: v=v1+v2 Gualtiero Giovanazzi [email protected] Vector v2 directed perpendicularly to the plane is the component of the Force (or acceleration) of gravity that holds the body to the plane. It is indicated by vperp v perp = v · cos(a) In this way we find the force that pulls the body downwards. This is vector v1 Lecture notes of Physics The Vectors Decomposition of a vector: Examples 3rd How changes the velocity vector for a cannon-ball? b2 a2 v v a c2 b1 a1 We choose 3 positions and we study the velocity vector v v We can recognize two particular direction: horizontal shifting and vertical shifting c1 Therefore we draw a Cartesian system of reference with the AXES horizontal and vertical We can decompose v on directions 1, 2 for find 2 vectors, v1 v2, whose sum is equal to v: v = v1 + v2 We can repeat this for every point of trajectory. The vectors v1 v2 are the velocity with there the cannon-ball moves in horizontal (v1) and vertical (v2) Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected] So we find the force that pulls down the body. It is the vector v1 Lecture notes of Physics Return The Vectors • Proof d = a b d = a b -b Difference between Vectors a - a b + -b b d = a b a d = a b Liceo Scientifico ‘Rainerum’d WeEuropeo can translate Gualtiero Giovanazzi [email protected] We can use only the addition of vectors and multiplication with a number We can use the parallelogram law The vector d, difference a-b, is a vector that start from the arrow of b and arrives at the arrow of a Lecture notes of Physics The Vectors The End Liceo Scientifico Europeo ‘Rainerum’ Gualtiero Giovanazzi [email protected]
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