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
u  2i  3 j , v  i  5 j  k , w  4 j  2k
a) Compute w  (u  2 v ) .
b) Compute || v  u || .
c) Compute (w·v )u .
d) Find a vector that is in the direction of the vector u , whose magnitude is that of vector v .
Problem 2: Plane geometry (2D).
Let s  4,1  and let t  1, 1  .
a) Show a sketch of the vector sum s  2t , illustrating the parallelogram method of addition. Make
it accurate (use graph paper).
b) Calculate the resultant s  2t (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 s and t .
d) Compute projs t and orth s t . Add them to your sketch.
e) Find the area of the parallelogram with sides s and t .
Problem 3: Space geometry (3D).
Consider the points P  (0, 0, 0) , Q  (4, 3, 0) , R  (0, 2, 6) , S  (7,3, 2) and form vectors PQ ,
PR etc. as needed to compute the following:
a) The magnitude, direction cosines and angles for the vector PS .
b) The area of the triangle PQR .
c) The volume of the parallelpiped with adjacent edges PQ , PR , and PS .
Problem 4: Orthogonality.
a) Give an example of a vector orthogonal to the vector v  5, 4  .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 v  1,5, 4  .How many such vectors
are there? How are they all related to each other?
c) Find a unit vector orthogonal to both the vectors v  3i  2 j  k and w  j  2k .
Problem 5: Applications:
a) A bag of cement of weight 325 N hangs in equilibrium from three wires as shown in the figure
below. Two of the wires make angles 1  60.0 and 2  25.0 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 T1 , T2 , and T3 in the wires.
b) The picture below shows a wrench with force F  56 lb applied to the end at an angle  .
Assume  is allowed to vary (so don’t assume that F must be directed downwards).
i.
ii.
iii.
Write an expression for the magnitude of the torque produced by applying force F . This
expression should be in terms of a variable  . Note: the 30 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   45 .
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 v in R n , c and d scalar,
(c  d ) v  cv  dv
b) Prove that the vector w || u || v  || v || u bisects the angle between u and v .
Here’s a few hints to get you going…
One way to show is to let  be the angle between u and w , and let  be the angle between v
and w . The goal is to show that    …which is implied if you can show cos   cos  . Use
the dot product/angle formula for each angle, sub in the expression for w , 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 u and v are orthogonal vectors in R3 . Prove that
|| u  v |||| u |||| v ||
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
u  3i  2 j  k
a) Compute  2u  v   w .
b) Compute || u  ( v  3w ) || .
v  5i  k
w  i  j  k