Transcription

1 ector ition 1. Scalar: physical quantity having only magnitue but no irection is calle a scalar. eg: Time, mass, istance, spee, electric charge, etc.. ector: physical quantity having both magnitue an irection an which obeys the laws of vector aition is calle a vector quantity. eg: Displacement, velocity, acceleration, intensity of electric fiel, etc. 3. Surface area can be treate both as a scalar an a vector. is magnitue of surface area which is a scalar. If ˆn is a unit vector normal to the surface, we can write n ˆ as a vector. 4. Electric current an velocity of light are not vectors even though they have irection since they o not obey the laws of aition. 5. vector quantity which has irection by its nature is calle a polar vector. Ex: velocity. 6. vector quantity which has irection by a convention is calle a pseuo (or) axial (or) non-polar vector. The irection of pseuo vector can be known from right han thumb rule. Ex: ngular velocity. 7. Equal vectors: ectors having same magnitue an which have same irection are calle equal vectors. Their corresponing components are equal. 8. Negative vectors: vector which has the same magnitue as that of another an which is opposite in irection is calle a negative vector. 9. Null ector (Zero ectors): vector whose magnitue is zero an which has no specific irection is calle a null vector. e.g. 1) The cross prouct of two parallel vectors is a null vector. ) The ifference of two equal vectors is a null vector. 10. Unit vector: It is a vector whose magnitue is unity. unit vector parallel to a given vector. If is a vector, the unit vector in the irection of is written as ˆ. î, ĵ ankˆ are units vectors along x, y an z axis. 11. osition vector: The position of a particle is escribe by a position vector which is rawn from the origin of a reference frame.the position vector of a particle in space is given by O r xi + yj + zk. O

2 Its magnitue is given by r x + y + z Unit vector of r is given by r xi + yj + zk r r x + y + z ition of ectors 1. esultant can be foun by using a) Triangle law of vectors b) arallelogram law of vectors c) olygon law of vectors 13. Triangle law: If two vectors are represente in magnitue an irection by the two sies of a triangle taken in orer, then the thir sie taken in the reverse orer represents their sum or resultant both in magnitue an irection. 14. arallelogram law If two vectors an Q are represente by the two sies of a parallelogram rawn from a point, then their resultant is represente in magnitue an irection by the iagonal of the parallelogram passing through that point. Tan + Q + Q cos Q sin α + Q cos an sin tan β Q + cos 15. If the resultant of an Q makes an angle α with an β with Q an if > Q thenα < β. 16. For two vectors an Q, max + Q an min Q 17. If two vectors an Q have equal magnitues x, then x cos 18. ectors aition obeys a) Commutative law: + + b) ssociative law: + ( + C) ( + ) + C c) Distributive law: m( ) m + m + where m is a scalar. 19. olygon law: If a number of vectors are represente by the sies of a polygon taken in the same orer, the resultant is represente by the closing sie of the polygon taken in the reverse orer. Q β α

3 0. esolution of a vector Consier a vector represente along O in two co-orinate system, which makes an Y angle with X-axis. i + j an x x Cos x Cos y Sin y Sin y x y + If the vector makes an angle α with X-axis, β with Y axis an γ with Z-axis Then i + j + k an x y z x y z + + Cos x ; Cos y z α l β m an Cosγ m Cosα + Cos β + Cos γ 1 (or) n l + m + n 1 Sin α + Sin β + Sin γ (Law of cosines) 1. If l 1, m1, n1 an l m n the irection cosines of two vectors an is the angle between them then cos ll 1 + m 1 m + n 1 n.. Component of a vector is a vector. 3. If vectors î + ĵ kˆ x y + K where K is a scalar. Equilibrium z ˆ i ˆ j k ˆ an x + y + z are parallel, then x x y z y z an 4. Equilibrium is the state of a boy in which there is no acceleration i.e., net force acting on a boy is zero. 5. The forces whose lines of action pass through a common point are calle concurrent forces. 6. esultant force is the single force which prouces the same effect as a given system of forces acting simultaneously. 7. force which when acting along with a given system of forces prouces equilibrium is calle the equilibrant. y j O x xi y X

4 8. esultant an equilibrant have equal magnitue an opposite irection. They act along the same line an they are themselves in equilibrium. 9. Triangle law of forces: If a boy is in equilibrium uner the action of three coplanar forces, then these forces can be represente in magnitue as well as irection by the three sies of a triangle taken in orer. Where p, q, r are sies of a triangle., Q, are coplanar vectors. p q r Q 30. Lami s theorem: When three coplanar forces, Q an keep a boy in equilibrium, then 1 3 n Q sinα sinβ sin γ. 31. If an 1 3 n, then the ajacent vectors 360. are incline to each other at an angle N π or N 3. N forces each of magnitue F are acting on a point an angle between any two ajacent forces is, then resultant force F resultant 33. oy ulle Horizontally N Fsin sin( / ) The horizontal force require to pull a suspene boy through an angle with the vertical is given by T sin F an T cos mg ( ) T F + mg F Tan mg F mg Tan Motion of the boat in a river 34. Let be the velocity of the boat an the velocity of the river.. γ β α l -x l x T mg r F p q Q Q 1. The time taken by the boat to go from to an to in still water T. If the river is flowing

5 t own + T t + tu 3. ( ) Sin t an t up sin cos Time taken by the boat to cross the river is t Sin o If 90, then t is minimum.i.e. the boat can cross the river in a shortest time if it moves along. 35. Shortest ath Sin n Or esultant velocity 1 Sin Sin with stream Time taken to cross the river 36. Shortest time esultant velocity + Time taken to cross the river is x lso Tan x 37. Subtraction of two vectors t t a) If an Q are two vectors, then Q is efine as + ( Q) ofq. If Q, then + Q QCos In the parallelogram OMLN, the iagonal OL represents represents C x C net + where Q is the negative vector + an the iagonal NM

6 N L b) subtraction of vectors oes not obey commutative law c) subtraction of vectors oes not obey ssociative law ( C) ( ) C O M. ) Subtraction of vectors obeys istributive law m ( ) m m 38. For two equal vectors x Sin