Fall 2010 Math 152
2
Week in Review XIV
Section 13.4
Key Concepts:
courtesy: David J. Manuel
1. Polar to Cartesian: x = r cos θ, y = r sin θ
(covering 11.3 and 13.4)
2. Cartesian to Polar: r2 = x2 + y 2 , tan θ =
1
3. Graphing Polar Curves (use quadrants and
symmetry)
Section 11.3
Key Concepts:
1. Cross Product
(|a||b| sin θ)n
y
x
Examples:
definition:
a × b
=
1. Convert the following points to cartesian coordinates:
π
(a) 4,
3
√
π
(b) 2 2, −
4
4π
(c) 6, −
3
2. Computation formula:
a×b = (a2 b3 −a3 b2 )i+(a3 b1 −a1 b3 )j+(a1 b2 −
a2 b1 )k
3. Result: a vector orthogonal to the plane
containing a and b whose magnitude is
|a||b| sin θ (area of the parallelogram formed
by a and b)
2. Convert the following points to polar coordinates:
4. Scalar Triple Product = |a · (b × c)| = volume
of the parallelipiped formed with a, b and c
at one corner.
(a) (−3, 3)
√
(b) (−2 3, −2)
√
√
(c) ( 6, − 2)
Examples:
3. Sketch the graphs of the following polar equations:
1. Find a × b if a = h1, −1, 2i and b = j − 3k.
2. Find
a
vector
parallel
to
the
plane
containing
the
points
A(2, 0, 1), B(4, 3, −1) and C(3, 2, 3).
(a) r = 3 cos θ
(b) r2 = cos 2θ
1
(c) r = (1 − 2 sin θ)
2
(d) r = a(1 + cos θ)
3. Find the area of △ABC in #2
4. Find the volume of the parallelipiped
which has a corner formed by the vectors
h−4, 3, −1i, h2, 3, −2i, and h1, 2, 2i.
4. Convert r = 3 cos θ + 2 sin θ to cartesian and
sketch the graph.
5. Find a unit vector orthogonal to the vectors
−i + j + 3k and −2j + k.
6. A 0.75m pipe wrench is held to a connecting bolt at an angle of 30◦ with the x-y plane.
Because of space limitation, the force of 20N
must be applied straight down (i.e., perpendicular to the x-y plane). Find the magnitude
and direction of the torque.
1