MATH200, Section 107, Rachel Ollivier
Fall 2013
Review 1
On the common website, there is more material to help you review for the midterm.
Consider our usual system of coordinates with origin O = (0, 0, 0) and axes given by the unit
vectors ~i, ~j and ~k. Recall that we have ~i × ~j = ~k.
Problem 1. Let ~a = −~i + ~k and ~b = 2~i + 2~j.
1. Find the cosine of the angle between ~a and ~b.
2. What is the area of the parallelogram determined by ~a and ~b ?
3. Find two unit vectors orthogonal to both ~a and ~b.
4. We consider the point P = (1, 2, 3).
−→
(a) Give the coordinates of the point Q such that P Q = ~a.
−→
(b) Give the coordinates of the point M such that M P = ~b.
(c) Find the equation of through the points P , Q and M .
(d) Is the plane P1 parallel to the plane with equation 9x − 12y − 11z = 9 ? Justify your
answer.
5. Let P2 be the plane through the origin and with normal vector ~i + ~j + ~k.
(a) What is the equation of P2 ?
(b) What is the cosine of the angle between P1 and P2 ?
(c) The intersection between P1 and P2 is a line which we denote by L. Give the parametric equation of L.
Hint : see Example 7 in Section 12.5.
Problem 2. 1. Why do we know that the planes P3 and P4 with respective equations
6x + 12y − 24z = 48 and x + 2y − 4z = 7 are parallel ?
2. We want to compute the distance between the point Ω = (3, −5, −2) and P3 .
(a) First we need a point on P3 : find k such that the point A = (0, 0, k) lies on the
plane P3 .
~ which is normal to P3 .
(b) Find a vector N
(c) Convince yourself that the distance from Ω to P3 is equal to the magnitude of the
−→
~ . Then compute it.
projection of AΩ on N
3. Find the distance between the parallel planes P3 and P4 .
1
2
4. Find the intersection between P4 and the line with parametric equation
x = 1 + 3t, y = −1 + t, z = 2t.
Problem 3. Let L be the line parametric equation
x = 1 − 3t, y = 1 + 2t, z = 7 + 2t
0
and L the line parametric equation
x = 1 + 3t, y = 1 − 5t, z = −t.
1. Which of the following points belong to L :
1
A = (−2, 3, 9), B = (1, 1, −1), C = (− , 2, 8) ?
2
0
2. Which of the following points belong to L :
2 1
A0 = (2, − , − ), B 0 = (1, 1, 0), C 0 = (1, 2, −8) ?
3 3
0
3. Are L and L parallel ? Justify your answer.
4. Prove that the line L00 with equation
2
2
~r(t) = (−2, 3, 9)+ < t, − t, − t >
3
3
is the same as L.
5. Is there a plane that is perpendicular to both L and L0 ?
6. Give a symmetric equation of L.
7. Show that L and L0 are skew.
8. Find the equation of a plane P that contains L and a plane P 0 that contains L0 such that
P and P 0 are parallel. Compute the distance between P and P 0 (recall that this is the
same as the distance from any point of P 0 to the plane P).
9. Deduce from the previous question the distance between L and L0 .
10. Observe that in fact we don’t really need both planes P and P 0 from Question 6 to compute
the distance between L and L0 . We could just
– find a plane P containing L and "parallel" to L0 .
– Compute the distance from any point of L0 to P.
11. Determine the distance from A to L.
12. Determine the distance d from B to L and find the coordinates of the unique point Ω on
L such that BΩ = d (it is the point on L that is the closest to B). Make a drawing !..