Friday, May 22, 2015

Vectors HOT Sheet 1

    The following forces act on a particle P: F1 = 2i + 3j - 5k, F2 = -5i + j + 3k, F3 = i - 2j + 4k, F4 = 4i - 3j - 2k, measured in Newtons. Find (a) the resultant of the forces, (b) the magnitude of the resultant.
    If A = 3i - j - 4k, B = -2i + 4j - 3k, C = i + 2j - k, find 2A - B + 3C, (b) |A + B + C|, (c) |3A - 2B + 4C|, (d) a unit vector parallel to 3A - 2B + 4C.
    The position vectors of points P and Q are given by rl = 2i + 3j - k, r2 = 4i - 3j + 2k. Determine PQ in terms of i, j, k and find its magnitude. 
    Show that the vectors A = 2i +4j – 3k and B = i + 2j + 2k are perpendicular to each other.

    Under what condition, the magnitude of the resultant vector of two equal vectors (a) may be zero, (b) may be equal to each other.
    An air plane is flying with a uniform speed of 100 kmph around the circumference of a circle. What will be the change in velocity during (i) a quarter round turn and (ii) half round turn?
    The sum and difference of two vectors a and b are mutually perpendicular to each other. Prove that the vectors are equal in magnitude.
    The sum and of two vectors are equal in magnitude such that |a + b| = |a - b|. Prove that the vectors a and b are perpendicular to each other.
    The resultant of two vectors a and b is perpendicular to the vector a and its magnitude is equal to half the magnitude of vector b. Find out the angle between a and b.

     A boat is rowed with a velocity of 6 km per hour straight across a river which flows at the rate of 2 km per hour. If its breadth be 300 metres, find how far down the river the boat will reach the opposite bank below the point at which it was originally directed.
     A man wishes to cross a river to an exactly opposite point on the other bank if he can pull his boat with twice the velocity of the current, find at what inclination to the current he must keep the boat pointed. 
     A stream runs with a velocity of 1.5 km per hour; find in what find in what direction, a swimmer, whose velocity is 2.5 km per hour, should start in order to cross the stream, perpendicularly. What direction should be taken in order to cross in the shortest time?
     A point possesses simultaneously velocities whose measures are 4, 3, 2 and 1. The angle between the first and second is 30deg, between the second and third 90deg, and between third and fourth 120deg; find their resultant. 

     One ship is sailing due east at the rate of 12 km per hour, and another ship is sailing due north at the rate of 16 km per hour; find the relative velocity of the second ship with respect to the first.
     A ship steams due west at the rate of 15 km per hour relative to the current which is flowing at the rate of 6 km per hour due south. What is the velocity relative to the ship, of a train going due north at the rate of 30 km per hour?
     A ship is sailing north at the rate of 4 metres per second; the current is taking it east at the rate of 3 metres per second, and a sailor is climbing a vertical pole at the rate of 2 metres per second; find the velocity of the sailor in space.
     In a tunnel, drops of water which are falling from the roof are noticed to pass the carriage window of a train in a direction making an angle tan-1(½) with the horizon, and they, are known to have a velocity of 730 cm per second. Neglecting the resistance of the air, find the velocity of the train.

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