Lecture 2: Multiplication of vector by real number ; Addition and subtraction of vector: graphical method ; Resolution of vectors

Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere ? Explain.

 A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector ? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector ?

 For any arbitrary motion in space, which of the following relations are true :
(a) v_average = (1/2) (v (t₁) + v (t₂))
(b) v_average = [ r(t₂) - r(t₁) ] / (t₂ – t₁)
(c) v (t) = v (0) + a t
(d) r (t) = r (0) + v (0) t + (1/2) a t²
(e) a_average = [ v (t₂) - v (t₁ ) ] / ( t₂ – t₁)
(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

 î and ĵ are unit vectors along x and y axis respectively. What is the magnitude and direction of the vectors î + ĵ, and î − ĵ ? What are the components of a vector A= 2î + 3ĵ along the directions of î + ĵ and î − ĵ? [You may use graphical method]

A particle starts from the origin at t = 0 s with a velocity of 10.0 ĵ m/s and moves in the x-y plane with a constant acceleration of (8.0 î+ 2.0 ĵ) m s^-2. (a) At what time is the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at that time? 
(b) What is the speed of the particle at the time ?

The position of a particle is given by

Given vectors a + b + c + d = 0, which of the following statements are correct:
(a) Vectors a, b, c, and d must each be a null vector,
(b) The magnitude of vectors (a + c) equals the magnitude of vectors(b+ d),
(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,
(d) Vectors b + c must lie in the plane of a and d if  a and d are not collinear, and in the line of a and d, if they are collinear?

Establish the following vector inequalities geometrically or otherwise:
(a) |a + b| ≤ |a| + |b|
(b) |a + b| ≥ ||a| − |b||
(c) |a − b| ≤ |a| + |b|
(d) |a − b| ≥ ||a| − |b||
When does the equality sign above apply?

 Pick out the only vector quantity in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

 State, for each of the following physical quantities, if it is a scalar or a vector:
volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.