GEOMETRY EXERCISES 1 Describe the regions of space given by the

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GEOMETRY EXERCISES 1 Describe the regions of space given by the Powered By Docstoc
					                                              GEOMETRY EXERCISES


1. Describe the regions of space given by the following vector equations. In each, r denotes the vector xi + yj + zk;
‘·’ and ∧ denote the scalar (dot) and vector (cross) product:


   • r∧ (i + j) = (i − j) ,
   • r · i = 1,
   • |r − i| = |r − j| ,
   • |r − i| = 1,
   • r · i = r · j = r · k,
   • r ∧ i = i.


2. Find the shortest distance between the lines
                                     x−1   y−3   z                             y−1
                                         =     =              and x = 2,           = z.
                                      2     3    2                              2
[Hint: parametrise the lines and write down the vector between two arbitrary points on the lines; then determine when
this vector is normal to both lines.]

3. Let Lθ denote the line through (a, b) making an angle θ with the x-axis. Show that Lθ is a tangent of the parabola
y = x2 if and only if
                                              tan2 θ − 4a tan θ + 4b = 0.
[Hint: parametrise Lθ as x = a + λ cos θ and y = b + λ sin θ and determine when Lθ meets the parabola precisely once.]

Show that the tangents from (a, b) to the parabola subtend an angle π/4 if and only if

                                                     1 + 24b + 16b2 = 16a2 .

[Hint: use the formula tan(θ1 − θ2 ) = (tan θ1 − tan θ2 )/(1 + tan θ1 tan θ2 ).]

Sketch the curve 1 + 24y + 16y 2 = 16x2 and the original parabola on the same axes.

4. What transformations of the xy-plane do the following matrices represent:
                    µ    ¶      µ        ¶µ     ¶            µ     ¶     µ       ¶µ     ¶
                       x          1 0        x                  x          2 0       x
               i)            7→                    ,     ii)          7→                  ,
                    µ y ¶       µ 0 −1       y
                                            ¶µ       ¶       µ y ¶       µ 0 1       y ¶µ     ¶
                       x          1/2 1/2        x              x          cos θ − sin θ    x
               iii)         7→                         , iv)         7→                         .
                       y          1/2 1/2        y              y          sin θ cos θ      y

Which, if any, of these transformations are invertible?

5. The cycloid is the curve given parametrically by the equations

                                x (t) = t − sin t,    and y (t) = 1 − cos t for 0 ≤ t ≤ 2π.

(a) Sketch the cycloid.

(b) Find the arc-length of the cycloid.

(c) Find the area bounded by the cycloid and the x-axis.

(d) Find the area of the surface of revolution generated by rotating the cycloid around the x-axis.

(e) Find the volume enclosed by the surface of revolution generated by rotating the cycloid around the x-axis.

				
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Description: GEOMETRY EXERCISES 1 Describe the regions of space given by the