MATH 205 - Lecture 1 - Problem Set
Turtle Problems
1. Compute the sum of the vectors ⟨−2, −1⟩ and ⟨5, 7⟩,
and illustrate the result geometrically.
2. Without calculating the angle explicitly determine whether
the angle between the two vectors
2⃗i − ⃗j + 7⃗k and 3⃗i − 2⃗j − ⃗k
is acute, obtuse, or 90 degrees.
3. Find the angle between the vectors ⟨2, 0, 2⟩ and ⟨0, 0, −3⟩.
There’s a bit of trig at the end, but if you remind yourself of the unit circle you should be able to get to the
answer without a calculator.
Dolphin Problems
4. Find the equation of the sphere that passes through
the origin and whose center is (1, 2, 3)?
5. Describe in words the region of the R3 represented by
the equation z 2 = 1.
6. Given ⃗a = 2⃗i − 4⃗j + 4⃗k and ⃗b = 2⃗j − ⃗k, compute
(a) 2⃗a + 3⃗b
(b) |⃗a|
(c) |⃗a − ⃗b|
Orca Problems*
∗
Orca Problems are optional.
7. Find the angle between a diagonal of a cube and the
diagonal of one of its faces.
You will need to use a calculator. Give your answer in
degrees, correct to one decimal place.
8. Show that the diagonals of an equilateral parallelogram
(all four sides are the same length) are orthogonal.