State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,

$\hat{i}$ and $\hat{j}$ are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors $\hat{i} + \hat{j}$ , and $\hat{i} - \hat{j}$ ? What are the components of a vector $A= 2 \hat{i} + 3 \hat{j}$ along the directions of $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$ ? [You may use graphical method]

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,

A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement,

For any arbitrary motion in space, which of the following relations are true : (d) $r (t) = r (0) + v (0) t + (1/2) a t^2$

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.

A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?

In a harbour, wind is blowing at the speed of $72 km/h$ and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of $51 km/h$ to the north, what is the direction of the flag on the mast of the boat ?

Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

For any arbitrary motion in space, which of the following relations are true : (b) $v_{average} = [r(t_2) - r(t_1) ] /(t_2 - t_1)$

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,

A man can swim with a speed of $4.0 km/h$ in still water. How long does he take to cross a river $1.0 km$ wide if the river flows steadily at $3.0 km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank ?

On an open ground, a motorist follows a track that turns to his left by an angle of $60^{\circ}$ after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.

A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,

Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless

Can you associate vectors with (a) the length of a wire bent into a loop,

The ceiling of a long hall is $25 m$ high. What is the maximum horizontal distance that a ball thrown with a speed of $40 m s^{-1}$ can go without hitting the ceiling of the hall ?

Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,

Establish the following vector inequalities geometrically or otherwise : (d) $|a-b| > ||a| - |b||$. When does the equality sign above apply?