On an open ground, a motorist follows a track that turns to his left by an angle of 600 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.
The path followed by the motorist is a regular hexagon with side 500 m, as shown in the given figure.
Let the motorist start from point P.
The motorist takes the third turn at S.
∴ Magnitude of displacement = PS = PV + VS = 500 + 500 = 1000 m
Total path length = PQ + QR + RS = 500 + 500 +500 = 1500 m
The motorist takes the sixth turn at point P, which is the starting point.
∴ Magnitude of displacement = 0
Total path length = PQ + QR + RS + ST + TU + UP
= 500 + 500 + 500 + 500 + 500 + 500
= 3000 m
The motorist takes the eight turn at point R.
∴ Magnitude of displacement = PR
Therefore, the magnitude of displacement is 866.03 m at an angle of 30° with PR.
This implies AC makes an angle 30° with the initial direction.
Total path length = 8 × 500 = 4000 m.
The aircraft takes 10s to go from A to B and AB subtends an angle of 30° at O.
In right angled triangle OAC,
AC = OC tan 15° = 3400 x 0·2679 = 911m
∴
Now, the speed of aircraft is,
Given a + b+ c + d = 0, which of the following statements
are correct :
(a) a, b, c, and d must each be a null vector,
(b) The magnitude of (a + c) equals the magnitude of
( b + d),
(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,
(d) 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 ?
a)
Incorrect
In order to make vectors a + b + c + d = 0, it is not necessary to have all the four given vectors to be null vectors. There are many other combinations which can give the sum zero.
(b)
Correct
a + b + c + d = 0
a + c = – (b + d)
Taking modulus on both the sides, we get,
| a + c | = | –(b + d)| = | b + d |
Hence, the magnitude of (a + c) is the same as the magnitude of (b + d).
(c)
Correct
a + b + c + d = 0
a = - (b + c + d)
Taking modulus both sides, we get,
| a | = | b + c + d |
| a | ≤ | a | + | b | + | c | ... (i)
Equation (i) shows that the magnitude of a is equal to or less than the sum of the magnitudes of b, c, and d.
Hence, the magnitude of vector a can never be greater than the sum of the magnitudes of b, c, and d.
(d)
Correct
For a + b + c + d = 0