Problem 1-1: Prove that three points x, y, z Rn lie on a line if and only
if there exist scalars t1 , t2 , t3 , not all zero, such that
t1 + t2 + t3 = 0
t1 x + t2 y + t3 z = 0.
(1)
(2)
First we prove the only if: Suppose that x, y, and z all lie on a line
Extra problem: Show there do not exist nonzero vectors x, y, z R2 which
are pairwise orthogonal, i.e., x is orthogonal to y, y is orthogonal to z, and
z is orthognal to x.
Let x = (x1 , x2 ), y = (y1 , y2 ), and z = (z1 , z2 ). The orthogonality assumptio
Problem 1-4: Consider a triangle with vertices x, y, z Rn . Let L1 denote
the line determined by x and the midpoint (y + z)/2, L2 the line determined
by y and (x + z)/2, and L3 the line determined by z and (x + y)/2. Show
that these lines all meet in the
Problem 1-15: Consider two diagonals of faces of a cube which intersect at
a vertex. Compute the angle between them.
We refer to the gure in Problem 1-16; the vertices of the cube are
cfw_(1, 1, 1).
Take the vertex to be v = (1, 1, 1), and the diagonals t
Problem 1-33: Consider two open balls B(a, r) and B(b, s) in the case
where a b = r + s. Then the two balls are tangent at x. What is x equal
to in terms of a, b, r, and s?
The point x is on the line segment [a, b], thus x = (1 t)a + tb for some
t [0, 1].
Problem 2-1:
Let f : R Rm be a function and L Rm . Write out the correct
denition of
lim f (t) = L.
tt0 ,t=t0
Then prove that f is continuous at t0 if and only if
lim
tt0 ,t=t0
f (t) = f (t0 ).
The denition was given in class: For each > 0 there exists a
Problem 1-23: Consider a triangle with vertices x, y, z Rn and sides of
lengths
a= yz
b= xz
c= xy .
Prove that the angle bisectors of the three angles of the triangle are concurrent, intersecting at the point
p :=
ax + by + cz
.
a+b+c
It suces to show tha