METU DEPARTMENT OF MATHEMATICS
MA TH 115 AN AL YT IC GE O M E T R Y
FI N A L EX A M I N A T I O N
2018- 2 0 1 9 F a l l S e m e s t e r
S at u rd a y, Ja n ua r y 1 2th, 2 0 19
QUESTION 1 (15 points)
Given the lines ℓ1:{ 𝑥 =3𝑡 +1
𝑦 =4𝑡− 1
𝑧=𝑡+5 and ℓ2: (𝑥,𝑦,𝑧)=(𝑡+12 , 2𝑡+15 ,−3𝑡 +2). Determine whether ℓ1
and ℓ2 are coincident, intersect at a point, parallel or skew.
Direction vectors of given lines are 𝑢
1=(3,4,1) and 𝑢
2=(1,2,−3). Since 3
1≠4
2≠1
−3, these vectors are neither
parallel nor coincident, hence ℓ1 and ℓ2 either intersect at a point or they are skew. To determine whether they
intersect or not we have to check the following system of equations for a simultaneous solution:
3𝑡+1=𝑠+12
4𝑡−1=2𝑠+15
𝑡+5=−3𝑠+2.
By adding all equations side by side we get 8𝑡+5= 29 which gives 𝑡 =3 and substituting this value of t in the first
eqution we obtain s= −2. Since these values of 𝑡 and 𝑠 satisfy the last two equations as well, 𝑡=3, 𝑠 =−2 is a
solution of the system. It follows that the lines intersect at the point (10,11,8).
QUESTION 2 (10 points)
Find cosine of the angle between the planes 𝑥+2𝑦+2𝑧−3=0 and 2𝑥−3𝑦+6𝑧+7=0.
The angle between two planes is defined to be the angle between their normal vectors. Normal vectors of the given
planes are 𝑛
1=(1,2,2) and 𝑛
2=(2,3,6). Then cosine the angle 𝜃 between the given planes is
cos𝜃 = 𝑛
1.𝑛
1
|𝑛
1||𝑛
2|=8
3⋅7=8
21.
QUESTION 3 (25 points)
In the coordinate system 𝑂𝑥𝑦 let 𝒞 be the circle (𝑥−2)2+𝑦2=1 in plane 𝑧=0.
a) Sketch a rough graph and write an equation of the surface 𝒮𝑥 obtained by rotating 𝒞 around 𝑂𝑥
axis.
Rotation around 𝑂𝑥 axis yields the equation (𝑥−2)2+𝑦2+𝑧2=1 which represents the sphere with center at
(2,0,0) and radius 1.
b) Sketch a rough graph and write an equation of the surface 𝒮𝑦 obtained by rotating 𝒞 around 𝑂𝑦
axis.
Rotation around 𝑂𝑦 axis results in the equation (√𝑥2+𝑧2−2)2+𝑦2=1 which represents a torus..
c) Determine the intersection of 𝒮𝑥 and 𝒮𝑦.
𝒮𝑥∩𝒮𝑦= 𝒞 .
