MTH 277 Graded Assignment 1: Modules 1 through 4
Problem 1: Basic computations with vectors; given two vectors, you should be able to compute magnitudes, vector sums and scalar multiples, dot and cross products. Please do these computations by hand (although you can use Maple or Scilab to check).
For the vectors
, ,
a)Compute .
b)Compute .
c)Compute .
d)Find a vector that is in the direction of the vector , whose magnitude is that of vector .
Problem 2: Plane geometry (2D).
Let and let .
a)Show a sketch of the vector sum , illustrating the parallelogram method of addition. Make it accurate (use graph paper).
b)Calculate the resultant (it should agree with your sketch, of course). Express in both component form, and as a magnitude/angle pair where is the angle the resultant vector makes with the positive x-axis.
c)Use the dot product formula to calculate the angle between the vectors and .
d)Compute and . Add them to your sketch.
e)Find the area of the parallelogram with sides and .
Problem 3: Space geometry (3D).
Consider the points , , , and form vectors , etc. as needed to compute the following:
a)The magnitude, direction cosines and angles for the vector .
b)The area of the triangle .
c)The volume of the parallelpiped with adjacent edges , , and .
Problem 4:Orthogonality.
a)Give an example of a vector orthogonal to the vector .How many such vectors are there? How are they all related to each other?
b)Give an example of a vector orthogonal to the vector .How many such vectors are there? How are they all related to each other?
c)Find a unit vector orthogonal to both the vectors and .
Problem 5: Applications:
a)A bag of cement of weight N hangs in equilibrium from three wires as shown in the figure below. Two of the wires make angles and with the horizontal (yes, that is meant to be horizontal – the scan was a little askew). Assuming the system is in equilibrium, find the tensions , , and in the wires.
b)The picture below shows a wrench with force lb applied to the end at an angle . Assume is allowed to vary (so don’t assume that must be directed downwards).
- Write an expression for the magnitude of the torque produced by applying force . This expression should be in terms of a variable . Note: the on the wrench isn’t relevant – the only angle that puts in an appearance in the expression for magnitude of torque is the angle between the force and the lever arm.
- Calculate the torque on the wrench when .
- What angle would maximize the torque on the wrench? What is the value of the maximum torque?
Problem 6: Theory- prove the following:
a)For a vector in , and scalar,
b)Prove that the vector bisects the angle between and. Here’s a few hints to get you going…
One way to show is to let be the angle between and, and let be the angle between and. The goal is to show that …which is implied if you can show . Use the dot product/angle formula for each angle, sub in the expression for , and see what comes out.
There’s another way you can show it based on the geometry as well. Whatever approach makes sense to you is fine…
c)Suppose that and are orthogonal vectors in . Prove that
Problem 7: More vector algebra. Compute each of the following by hand, then, check your work using SciLab (or Maple, if you prefer). Attach a printout of your SciLab calculations.
Let
a)Compute.
b)Compute .