If \(\overrightarrow { { a }_{ 1 } } \) and \(\overrightarrow { { a }_{ 2 } } \) are two non collinear unit vectors and if \(\left| \overrightarrow { { a }_{ 1 } } +\overrightarrow { { a }_{ 2 } } \right| \) =\(\sqrt{3}\), then the value of \(\left( \overrightarrow { { a }_{ 1 } } -\overrightarrow { { a }_{ 2 } } \right) .\left( 2\overrightarrow { { a }_{ 1 } } +\overrightarrow { { a }_{ 2 } } \right) \) is

The sum of magnitudes of two forces acting at a point is 18 units and the magnitude of their resultant is 12 units. The resultant is at 90° with the force of the smaller magnitude. The magnitude of the individual forces is

From the top of a tower of height 40 m, a ball is projected upwards with a speed of 20 m/ s at an angle of elevation of 30°. The ratio of the total time taken by the ball to hit the ground to its time of flight (time taken to come back to the same elevation) is (Take g = 10 m/s²)

A boy aims a gun at a target from a point, at a horizontal distance of 100 m. if the gun can impart a horizontal velocity of 500 ms^-1 to the bullet, the height above the target where he must aim his gun, in order to hit it is (Take g = 10 ms=^-2)

At the top of the trajectory of a projectile, the directions of its velocity and accelerations are

What is the angle made by vector, A = \(2\hat { i } +2\hat { j } \) with x-axis?

The magnitude of vectors A, B and C are 12,5 and 13 units respectively and A + B = C, find the angle between A and B.

What are the minimum  number of forces which are numerically equal whose vector sum can be zero?

Under what condition the three vectors cannot give zero resultant?

Can the walking on a road be an example of resolution of vectors?

Explain the property of two vectors A and B if \(|\mathbf{A}+\mathbf{B}|=|\mathbf{A}-\mathbf{B}|\)

The dot product of two vectors vanished when vectors are orthogonal and has maximum value when vectors are parallel to each other.Explain

The angle between vector A and B is 60⁰  .What is the ratio of A.B and \(\left| A\times B \right| \) ?

Two bodies are projected at an angle \(\theta \) and to the(90⁰-\(\theta \)) to the horizontal with same speed. Find the ratio of their time of flight.

What is the angle of projection at which the h_max and range are equal?

There are two displacement vectors,one of the magnitude 3m and the other of 4m.How would two vectors be added so that the magnitude of the resultant vector be (i) 7 m (ii) 1 m  and (iii) 5 m.

A bullet P is fired from a gun when the angle of elevation of the gun is 30⁰. Another bullet Q is fired from the gun when the angle of elevation is 60⁰. The vertical height attained in the second case is x times the vertical height attained in the first case. What is the value of x?

Briefly disCI/5S subtraction of vectors.

Calculate the angular speed of the seconds hand of a clock. If the length of the seconds hand is 4 cm, calculate the speed of the tip of the seconds hand.

What would be the effect on a vector if all its components are reversed in direction?

A motorboat is racing towards north at 25 km/h and the water current in that region is 10 km/h in the direction of 60° east of south. Find the resultant velocity of the boat.

A cricket ball us thrown at a speed of 28\({ ms }^{ -1 }\)in a direction \({ 30 }^{ \circ }\)above the horizontal. 
(i) the maximum height
(ii) the time taken by the ball to return to the same level and
(iii) the distance from the thrower to the point where the baU returns to the same level.

A ball is thrown from a point in level with and at a horizontal distance r from the top of a tower of height H. How must the speed and angle of projection of the ball be related to r in order that the ball may just go grazing past the top edge of the tower? At what horizontal distance x from the foot of the tower does the ball hit the ground? For a given speed of projection, obtain an equation for finding the angle of projection so that x is at a minimum.