L2 - UE MAT334
Exercise sheet n◦ 1
Basic geometry of IRn for n = 2 or 3
We suppose that the space IRn is provided with the usual Euclidean scalar product.
1. Plot the following sets in IR2 .
— D1 = {(x, y)| 0 ≤ x + y ≤ 1, y > 0}
— D2 = {(x, y)| x + y + 1 ≥ 0, x ≤ 0, y ≤ 0}
— D3 = {(x, y)| |x + y| ≤ 1, |x − y| ≤ 1}
— D4 = {(x, y)| x2 + y 2 ≤ 4}
— D5 = {(x, y)| x2 + y 2 > 1}
— D6 = {(x, y)| x2 + y 2 − 2x − 4y ≤ 4}
— D7 = {(x, y)| x2 + y 2 ≤ 1, x > 0, y ≥ 0}
— D8 = {(x, y)| x2 + y 2 ≤ 4, x + y ≥ 0}
— D9 = {(x, y)| x2 + y 2 ≤ 1, x + y ≥ 1}
— D10 = {(x, y)| x2 + y 2 ≤ 1 or x + y ≥ 1}
2. Consider four points A, B, C, D in the space IRn . Compute the following expression
−−→ −−→ −→ −−→ −−→ −−→
AB · CD + AC · DB + AD · BC.
3. Consider four points A, B, C, D in the space IR3 . Compute
−−→ −→ −−→ −−→
AB × AC − BC × BA
and
−→ −−→ −−→ −−→ −−→ −−→ −−→ −−→
CA × CB − DA × DB − DB × DC − DC × DA.
→ →
→
4. Let u , v et w be three vectors in the space IR3 . We recall the following identity for the double
vector product
→
→
→
→
→ →
→
→ →
u ×( v × w) = ( u · w) v −( u · v ) w (note that the order is important !)
Compute
→
→
→
→
→
→
→
→
→
u ×( v × w)+ v ×(w × u )+ w ×( u × v ).
Reminder
We recall that the orthogonal projection of a point P0 onto the line D is the point H on D
such that the distance d(P0 , H) is minimal with respect to all the points of the line. The value
1
of this minimum is called the distance from P0 to the line D (see the figure below).
P0
H
d
D
P1
5. Let D be the line in IR2 given by y = 2x − 1 and let A be the point (1, 2).
1. Compute the square of the distance between the points A and M (x, 2x − 1) of the line, as
a function of x. Show that this function has a minimum and deduce the distance d(A, D)
from A to the line D. Compute the coordinates of the projection H of A onto D.
2. Give a direction vector of D. Let B be point (−1, 1). Compute the distance from B to
the line D.
6. Let P1 (3, 1, −2) and P2 (−1, 2, 4) be two points in IR3 .
→
1. Find the equation of the line D containing P1 and P2 and a direction vector v of norm
1 of this line.
2. Compute the distance from P0 (1, 3, −1) to the line D.
3. Determine the coordinates (x, y, z) of the orthogonal projection H of P0 onto the line D.
7. Let P1 and P2 be the the planes of equations
P1 : 2x + 3y + z − 4 = 0;
P2 : 3x − y − 3z − 2 = 0.
1. Show that the two planes are orthogonal.
2. Compute the distance from the origin O to the plan P1 , to the plane P2 , and to the line
D = P1 ∩ P2 .
3. Consider the sphere S given by the equation x2 + y 2 + z 2 − 2x − 2y − 2z + 2 = 0. Is
the intersection between the plane P1 and the sphere S nonempty ? And the intersection
between the line D and the sphere S ?
2
P0
n
P2
H
P
P3
P1
Polar, cylindrical and spherical coordinates
8. Using polar coordinates, what does each of the two equations r = constant and θ = constant
represent, respectively ?
9. Consider cylindrical coordinates. What does each of the three equations r = constant,
θ = constant and z = constant represent, respectively ? And what does the set given by the
intersection of the conditions r = constant and θ = constant represent ? Answer the same
question for the intersection between r = constant and z = constant ?
10. Consider now spherical coordinates. What does each of the three equations r = constant,
θ = constant and φ = constant represent, respectively ? And what does the set given by the
intersection of the conditions r = constant and θ = constant represent ? Answer the same
question for the intersection between r = constant and φ = constant ?
11. Let C be the circle of equation (x − 1)2 + y 2 = 1. Give the equation determining C in polar
coordinates.
p
12. Let C be the cone in IR3 defined as z = x2 + y 2 in Cartesian coordinates. Give the equation
of C in cylindrical coordinates.
3
Homework
→
13. Let D1 be the line containing the point M1 and of direction vector u 1 , and let D2 be the
→
line containing the point M2 and of direction vector u 2 . The distance d(D1 , D2 ) between D1
and D2 is defined as the minimum of the distances d(M, M 0 ) between any two points M ∈ D2
→
→
and M 0 ∈ D2 . We assume that u 1 et u 2 are linearly independent.
→
→
1. Let P be the plane containing the point M1 and parallel to u 1 and u 2 . Give the equation
of P, and show that for every point M 0 of the line D2 we have that d(P, M 0 ) = d(P, M2 ).
Deduce that D2 is parallel to the plane P.
→ →
−−−−→
2. Compute the volume of the parallelepiped of sides u 1 , u 2 and M1 M2 . Deduce the distance
from the point M2 to the plane P, and the distance between the lines D1 and D2 .
14. A sailor is planning a journey from Brest to New-York by travelling along an arc of a circle
over the Atlantic Ocean. Compute the length of this arc Brest-New-York, if the Earth radius is
R = 6400km and the geographic coordinates of the cities are
Brest : latitude 48o 500 N , longitude 0o
New-York : latitude 40o 400 N , longitude 73o 500 W .
→
→
→
Hint : Let a 1 and a 2 be two vectors of the same norm. Show that the angle θ12 between a 1
→
and a 2 satisfies the identity
cos θ12 = sin θ1 sin θ2 cos(φ1 − φ2 ) + cos θ1 cos θ2 ,
→
→
where θ1 , θ2 , φ1 and φ2 are the angles describing the corresponding vectors a 1 and a 2 in spherical
coordinates.
The velocity and the acceleration
15. Consider a material point described by the smooth parameterized curve given by
x(t) = sin t,
y(t) = 1 − cos2 t,
where t ∈ IR indicates the time.
1. Plot the trajectory of the point and show that the motion is in fact periodic. Determine
the (smallest strictly positive) period.
2. Compute the velocity and determine the values of t where it vanishes. Compute also the
values of t where the speed (i.e. the norm of the velocity) is maximal.
4