VECTORSA physical quantity is said to a vector quantity if in addition to magnitudehas a specified directionobeys the law of parallelogram of addition, and this addition is cumulative, A vector does not change when-1. It is displaced parallel to itself anywhere in space.2. It is rotated through an angle which is an integral multiple of 2.A vector is represented by an arrow. Its length represents the magnitude and arrow head represents direction.NOTEA vector and the magnitude of a vector do not depend on the orientation of the axes if different sets of co-ordinate axes having different orientations are used.The quantities which are not completely specified by magnitude and direction are called tensors. Moment of inertia, radius of gyration, pressure, stress, density, refractive index, modulus of elasticity, dielectric constant, wave velocity and conductivity etc, are such quantities. Zero Vector or Null VectorIt is a vector whose magnitude is zero and direction indeterminate. A zero vector can be obtained by-multiplying a vector by zero, i.e., 0 () = by adding a vector and its negative vector, i.e., + () = NOTEIf a zero vector is added to or subtracted from a given vector, the vector does not change, i.e., + = and = If a zero vector is multiplied by non zero real number, a zero vector results, i.e., n () = If and are two non zero real numbers, then = holds only when = = Unit VectorA vector whose magnitude is equal to unity is called unit vector. Its direction is along the vector in whose direction it is defined. A unit vector in the direction of is (A cap). Thus, = where A is the magnitude of .= unit vector along X-axis, = unit vector along Y-axis and = unit vector along Z-axisEqual or Identical VectorsVectors having the same magnitude and acting in the same direction are called equal or identical vectors.Parallel VectorsAll the vectors having the same direction are called parallel vectors. Such vectors may be divided into two categories- (a) Like parallel vectors (same direction) (b) Unlike parallel vectors (opposite directions).Exercise 1What is the unit vector in the direction of = ?Exercise 2Vector 0.4 + 0.8 + c represents a unit vector. What is the value of c?Exercise 3 What is the unit vector parallel to the resultant of the vectors 4 + 3 + and + 3 3?Addition of Two Vectors Triangle Method of Vector Addition: If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then the resultant is represented in direction and magnitude by the third side of the triangle taken in opposite sense. One of the vectors () is drawn first and then from its head the second vector () is drawn. The vector obtained by joining the tail of with the head of gives the vector + (=) as shown in the figure. Parallelogram Method of Vector Addition: If two vectors acting at a point simultaneously are represented by two adjacent sides of a parallelogram in magnitude and direction, then their resultant in magnitude and direction is given by the diagonal of parallelogram passing through that point. This is shown in the figure. If a given number of vectors are represented in magnitude and direction by the sides of a polygon taken in order, then their resultant in magnitude and direction is given by the side completing the polygon in opposite order.Resultant of Two VectorsIfand are two vectors inclined with each other at an angle and their resultant is (= +), the magnitude of their resultant is given by R = And if makes an angle with , then, tan = NOTEThe maximum and minimum values of the resultant are given as R = P + Q (when = 0, i.e. vectors point in the same direction)and R = P – Q (when = 180, i.e. vectors point in the opposite directions)If the two vectors are perpendicular to each other, i.e., = 90, the resultant is given by R = and tan = If two vectors have equal magnitude (i.e., P = Q), then the resultant of their vector sum is given by R = P = P = 2P Similarly, the resultant of their vector subtraction is given by R = P ( is replaced by 180 ) = P = P = 2P If two vectors are in a plane but the third vector is in a different plane, then their resultant can never be zero.Minimum number of co-planar but unequal vectors, which can have their resultant zero is 3. For equal vectors this number is 2.Minimum number of non-coplanar vectors, which can have their resultant zero is 4.Exercise 4If two forces are equal and their resultant is also equal to one of them, then what is the angle between the forces? Exercise 5Following sets of three forces act on a body. In which cases the resultant cannot be zero? (a) 20 N, 20 N, 20 N (b) 20 N, 20 N, 40 N (c) 20 N, 40 N, 70 N (d) 20 N, 40 N, 60 N (e) 10 N, 20 N, 40 NExercise 6Of the given pairs of forces, the resultant of which cannot be 6 N?(a) 2 N & 3 N (b) 2 N & 6 N (c) 4 N & 4 N (d) 2 N & 9 N (e) 6 N & 6 NExercise 7A car is moving in a circular path of radius r with a speed v. What is the change in velocity in moving from P to Q?Resolution of a VectorA vector can be decomposed into an infinite number of pairs of components in two-dimensional space as a given vector can be the diagonal of infinite number of parallelograms. But if a vector is resolved in two components at right angles to each other (called the rectangular components), then only one set is possible.Let we find the components of along X-axis and Y-axis. If makes an angle with the X-axis, then the X- and Y-components are given by R = R cos --------- (1)And R = R sin --------- (2) If and are the vectors having magnitudes R and R respectively, then = R and = Rand from vector addition, = + = R + RNOTEIf components are given then the magnitude of a vector can be obtained by squaring and adding the equations (1) and (2) and we get R = The direction of can be obtained by dividing equation (2) by equation (1) astan = If R = R, then R = 0 or if R = R, then R = 0It means that the component of a vector perpendicular to itself is always zero.If vector is resolved into three mutually perpendicular components, then = R + R + and the magnitude is given by R = If vector makes angles, and with X, Y and Z-axes respectively, then cosines of these angles are called direction cosines of vector . These are given as cos = , cos = , cos = Also,cos + cos + cos = 1 Law of Triangle of ForcesIf three forces (say, , and ) act on a particle such that they can be represented in magnitude and direction by the adjacent sides of a triangle taken in cyclic order, then the particle is said to be in equilibrium. The converse of this law is also true, i.e., if a point is in equilibrium when acted upon by three forces, then they can be represented in direction and magnitude by the adjacent sides of a triangle taken in cyclic order. Lami’s TheoremIf a particle is in equilibrium under the action of three forces (say, , and as shown), then each force is proportional to sine of the angle between the other two, i.e., = = Product of a Vector and a ScalarIf a vector is multiplied by a scalar, the resulting quantity is a vector quantity having the same or opposite direction as that of original vector depending upon whether the scalar is positive or negative. For example, n and n have the same magnitude nX but opposite directions.If n is a real number or a dimensionless scalar, then the units of n and will be the same.Scalar or Dot ProductScalar or dot product of two vectors, and , inclined at an angle is defined as, . = PQ cos NOTE. = P Q cos = P (Q cos ) = Magnitude of component of along = Q (P cos ) = Magnitude of component of along . = . = . = 1. = . = . = 0The examples of scalar product are work W, power P, magnetic flux, etc.Exercise 8An object constrained to move in y direction is subjected to a force = () N. What is the work done by this force in moving the object through a distance 20 m along y-axis?Exercise 9 A force of 2 + 3 + 4 N acts on a body for 4 sec and produces a displacement of 3 + 4 + 5 m. What is the power used?Vector or Cross ProductVector or cross product of two vectors, and , inclined at an angle is defined as, = PQ sin , where is a unit vector perpendicular to both and .Examples of vector product are torque, angular momentum, force on a charge moving in an electric field, etc.NOTEIf = P + P + P and = Q + Q + Q, then,(i) . = P Q + P Q + P Q (ii) = = = = 0 = = = 1 The magnitude of (i.e., PQ sin ) is equal to the area of the parallelogram made by and . Exercise 10What is the torque of the force = () acting at point ( 2, 3, 4) about the point (1, 2, 3)? Exercise 11What is the area of a parallelogram formed by adjacent sides as the vectors 3 + 2 and 2 4?Angle Between Two VectorsAngle between two vectors, and can be calculated using any of the following relations,1. cos = 2. sin = 3. tan = NOTEVector + makes an angle of 45 with X- and Y-axis. With Z-axis, it makes angle of 90. Vector + + makes an angle of 54.74 with each of X-, Y- and Z-axes.Exercise 12Find the angle between the vectors and .Exercise 13Find the angle between the vectors 4 + 4 and 4 4.Relative VelocityIf two objects A and B are moving with velocities and , then The relative velocity of A with respect to B is = , The relative velocity of B with respect to A is = The magnitude of both and will be equal. It will be equal to the difference in magnitudes of and if they are moving in the same direction and it will be given by = = But their directions will be opposite to each other. The magnitude of both and will be equal to the sum of their magnitudes of and if they are moving in opposite directions and it will be given by = = + But their directions will again be opposite to each other. If the objects are moving at an angle with each other, then the relative velocity is calculated using the same method of vector subtraction (i.e., = ). The magnitude of relative velocity is given by (or ) = = If makes an angle with , then tan = = If the objects are moving at right angles to each other, then the magnitude of relative velocity is given by = And tan = Exercise 14A bird is flying towards south with a velocity 30 km/hr and a train is moving with a velocity 30 km/hr towards east. What is the velocity of the bird with respect to the train?Relative Velocity in Rain-Man System If rain drops fall vertically with velocity u and a man moves horizontally with velocity v, then the relative velocity of rain drops with respect to man is given by = The drops fall at an angle with the vertical given by tan = NOTEIt is at this angle that he should point his umbrella with the vertical to save himself from drops.Exercise 15 Rain is falling vertically downwards with a velocity 4 km/hr. A man moves on a straight road with a velocity 3 km/hr. What is the apparent velocity of the rain with respect to man and the apparent direction of the rain drops with the vertical?Relative Velocity in River-Boat (or Man) SystemIf a boat (or man) moving with velocity u in still water crosses a river flowing with a velocity v, thento cross the river in shortest time, the boat (or man) must start (at point A) at right angles to the direction of flow of water, i.e., toward a point exactly opposite on the other bank (pointB). To prove this, let the man starts in a direction at an angle with AB. In this situation, time taken to cross the river will be t = This time will be minimum when is maximum, i.e., = 1,i.e., = 0Thus, to cross the river in shortest time, the man (or boat) should swim at right angles to the direction of flow and this time is given by t = In this case, he will not reach the opposite point but some distance down the river (point C). The angle is given by tan = The distance BC is given by BC = v t’ = AB v/u to cross the river by a shortest path, he must actually reach the opposite end (point B). To do so, he must start towards a point up the river (point C). In this case, sin = To get the angle with the direction of flow, 90 must be added to. Exercise 16A boat moves with a speed of 5 km/hr with respect to water flowing with a speed of 3 km/hr. The river has a width of 1 km. What is the time taken by the boat to cross the river?Answers:1. () 2. 3. (3 + 6 2) 4. /3 radian 5. c, e 6. a, d 7. 2v sin 20 8. (a) 6 m, 3 m (b) m (c) 8. 20 J 9. 9.5 W 10. 4 4 N-m 11. 12. 13. 90 14. 30 km/hr S-W 15. 5 km/hr at an angle with the vertical 16. 15 minutes