MATH-203 Calculus of Multivariables and PDEs
Marks: 10
Instruction: Your assignment should be written neatly, and it will carry 2 marks. Due Date: 21 October,
2024. The assignment must be solved by all the students separately. Allocation of questions is based on the
last digit of your ID number.
Questions (1, 11, 13, 19) to be solved by the students who have ID ending at 0, 1
Questions (7, 17, 19, 23) to be solved by the students who have ID ending at 7, 8, 9.
Questions (2, 14, 16, 22) to be solved by the students who have ID ending at 2.
Questions (3, 9, 21, 23) to be solved by the students who have ID ending at 3.
Questions (4, 8, 12, 16) to be solved by the students who have ID ending at 4.
Questions (5, 10, 15, 20) to be solved by the students who have ID ending at 5.
Questions (6, 12, 18, 24) to be solved by the students who have ID ending at 6.
Question 1: In air navigation, direction are specified by measuring from the north in a clockwise direction. Suppose an airplane with an airspeed of 200mi/hr is flying in the direction 60o , and a wind is
blowing directly from the west at 40mi/hr. These velocities may be represented by vectors V and W is the
true course of the airplane relative to the ground, and the magnitude ||V+W|| is the ground speed of the
airplane. Approximate the ground speed to the nearest mile per hour and the true course to the nearest
degree.
Question 2: An airplane pilot wishes to mantian a true course in the direction 240o with ground speed of
400mi/hr when the wind is blowing directly north at 50mi/hr. Find the required air speed and compass
heading.
Question 3: A 150 kg crate is being dragged across a rough surface by a worker. The worker applies
a force vector F = (200, 100, 0)N, while the crate moves along a displacement vector d = (3, 1, 0)m. The
surface is inclined at an angle, causing the force and displacement vectors to have both horizontal and
vertical components. Calculate the work done.
Question 4: A 3-phase electrical system has a power line delivering voltage and current to a motor. The
voltage is represented by the vector V = (220, −110, 60)V, and the current flowing through it is given by the
vector I = (3, −2, 4)A. The system is operating in a non-ideal condition, resulting in complex power flows
in different directions. Determine the instantaneous power delivered to the motor.
Question 5: Two cranes are lifting a heavy object by applying forces at different angles. The first crane
exerts a force vector F1 = (50, 30, 10)kN, and the second crane applies a force vector F2 = (30, −15, 25)kN. To
ensure the safety of the lift, the engineer needs to find the angle between the two forces to evaluate their
combined effect. Calculate the angle between the two force vectors.
Question 6: A mechanical arm rotates a component using a force vector F = (100, 150, 0)N, applied at a
position vector r = (0.5, 0, 0) m from the pivot point. The system designer wants to calculate the torque
acting on the component around the pivot point using the cross product, but first, the component of the force
in the direction of the position vector needs to be found using the dot product. Determine the magnitude
of the component of F in the direction of r.
Question 7: A mechanical beam is subjected to two load forces F1 = (120, 80, 40)N and F2 = (30, −10, 50)N
acting at the same point. The engineer needs to find the projection of the first load force F1 onto F2 to analyze
how much of F1 is aligned in the direction of F2 . Calculate the projection of F1 onto F2 .
Question 8: A robotic arm in a factory is designed to pick and place objects. The arm is fixed at a
pivot point and a force vector F = (20, 15, 0)N is applied at a point r = (0.5, 0.3, 0)m from the pivot. The force
is applied at an angle, causing the arm to rotate around the pivot point. Calculate the torque generated
about the pivot
Question 9: An engineer is analyzing the effect of a force F = (40, −25, 15)N applied to a beam at a point
located by the position vector r = (3, 2, 0) m relative to a fixed point on the beam. To evaluate the bending
effect, find the moment of the force about the fixed point. This moment will help determine the bending
stress on the beam.
Question 10: A structural engineer is studying the geometry of a triangular support frame in a 3D space.
The triangle is formed by three vertices: A = (1, 2, 3), B = (4, 0, 6), and C = (5, 3, 1). The engineer needs to
calculate the area of this triangle to determine the load distribution on the structure. Use the cross product
of vectors AB and AC (where AB and AC are formed from points A, B, and C) to find the area of the triangle.
Question 11: An electrical engineer is studying the effect of a magnetic field on a current-carrying conductor. The wire is oriented along the vector L = (0.6, 0.8, 0)m, and it carries a current I = 15A. It is placed in a
uniform magnetic field B = (0, 0, 0.5)T. Calculate the magnetic force experienced by the wire.
Question 12: A component in a machine is subjected to a force F = (10, 20, 5)N at a position vector
r = (1, −2, 3)m relative to a fixed point. To determine how much of this force is perpendicular to the position
vector, find the component of F that is perpendicular to r using the cross product and calculate the resulting
vector.
Question 13: An object is acted upon by a force vector F = (30, 50, 70)N at a point P = (2, 1, −1)m. The
engineer needs to find the line of action of the force to understand how it affects the object’s rotation about
the origin. Determine the moment of this force about the origin using the cross product M = PF and describe
its direction.
Question 14: Find the equation of the plane passing through the points P(2, 3, 4), Q(5, 0, −2), and R(−1, 6, 3).
Question 15: Determine the equation of a plane through the points P1 (4, −1, 2), P2 (2, 5, −3), and P3 (−3, 2, 7).
Question 16: Find the parametric equations of a plane containing the points A(1, 2, 0), B(0, 3, 4), and
C(−2, −1, 3).
Question 17: Through the points P(−5, 3, 2), Q(1, −1, 4), and R(2, 0, −3), find the scalar equation of the plane.
Question 18: Given three points P1 (6, 2, −1), P2 (−4, 0, 3), and P3 (1, −3, 5), calculate the normal vector and
then find the equation of the plane.
Question 19: Find the parametric and scalar equations of the plane passing through points P(1, 2, 3),
Q(4, −1, 2), and R(0, 0, −3), and calculate the distance from a point S(3, 1, −2) to the plane.
Question 21: Find the Tangent, Normal and Binormal vectors of (Sin3t)(cost)i + (sin3t)(sint)j + tk, also
the curvature and torsion of the curve.
Question 22: Find the Tangent, Normal and Binormal vectors of (cost)i + (sint) j + (sin2t)k, also the curvature
and torsion of the curve.
Question 23: Find the Tangent, Normal and Binormal vectors of (4 + Sin20t)(cost)i + (4 + sin20t)(sint)j +
(cos20t)k, also the curvature and torsion of the curve.
Question 24: Find the Tangent, Normal and Binormal vectors of (t2 )i + (2sint) j + (2cost)k, also the curvature
and torsion of the curve.