Transcript Ishbel Saxton

Writing Enhancement for
Mathematics Undergraduates
Writing Enhancement for
Mathematics Undergraduates
1st year undergraduates
1½ hours per week
2 semesters, 10 hours per semester
Writing Enhancement for
Mathematics Undergraduates
Further Vector Methods - Distance Between Two Lines
A pair of two lines satisfy one of three conditions
1. They are parallel
2. The two lines intersect at a point
3. They are skew lines i.e. they are not parallel and they never meet.
The shortest distance between two skew lines will lie along an
imaginary
line which is perpendicular to both the two original lines. This imaginary
line joins the points on the two lines which are closest to each other. In
the diagram below, the lines are drawn as long, thin triangles to
emphasize
the three-dimensional nature e.g. the lines tapering away into the
distance.
There is NO crossing point in the right-part of the diagram.
Writing Enhancement for
Mathematics Undergraduates
Writing Enhancement for
Mathematics Undergraduates
The two lines are (without loss of generality, r1 = a + b and r2 = c + μd.
An algorithm for finding the two closest points and the distance between
them is as follows.
Step 1 :- Find n = b × d i.e. a vector perpendicular to both lines
Step 2 :- Let r2 = r1 + kn i.e. c + μd = a + b + kn i.e. a vector
from one
line to the other is perpendicular to both lines
Step 3 :- From the results of step 2, form three equations in , μ
and k.
(i.e. one equation from each of the x, y, and z-components)
Step 4 :- Solve the equations in step 3 to find , μ and k
Step 5 :- a + b and c + d are the closest points on the two lines
Step 6 :- Find the distance between the two points
Writing Enhancement for
Mathematics Undergraduates
A king wants his daughter to marry the smartest of three
extremely intelligent young princes, and so the king's wise
men have devised an intelligence test.
The princes are gathered into a room and seated, facing
one another, and are shown 2 black hats and 3 white hats.
They are blindfolded, and a hat is placed on each of their
heads, while the remaining hats are hidden in a different
room.
Writing Enhancement for
Mathematics Undergraduates
The king tells them that the first prince to deduce the colour of
his hat without removing it or looking at it will marry his
daughter. A wrong guess will mean death. The blindfolds are
then removed.
You are one of the princes. You see 2 white hats on the other
prince's heads. After some time you realize that the other
prince's are unable to deduce the colour of their hat, or are
unwilling to guess. What colour is your hat?
Note: You know that your competitors are very intelligent and
want nothing more than to marry the princess. You also know
that the king is a man of his word, and he has said that the test
is a fair test of intelligence and bravery.
Writing Enhancement for
Mathematics Undergraduates
If your hat was black, then the other two princes would
each see one white hat and one black hat. This would
not be fair as they would not be starting with the same
initial information as you and they would only have a fifty
per cent chance of guessing the colour of their hat.
Therefore, if the king is playing fair, which he must be
since he said the test was fair, your hat must be white
since each prince must be looking at two white hats.
Writing Enhancement for
Mathematics Undergraduates
English tutor from the University Language Centre
Graduate Teaching Assistant from the Maths Dept
Writing Enhancement for
Mathematics Undergraduates
Writing Enhancement for
Mathematics Undergraduates
problems
time – Wednesday afternoon
content – difficult to select