(b) Now nd the distance between the point
(2, 1, 5) and the line x = 3t 5, y = 1 t,
z = 4t + 7.
There is a nice way to do this that doesnt involve
Section 1.2 selected answers
calculus, but its not hard to do it using calculus.
Math 131 Multivariate Calcu
In order for g to be dened, there are two requirements. First, 4x2 y 2 z 2 has to be 0 so that its
square root is dened. Second, 4 x2 y 2 z 2
cannot be 0, since its in a denominator. Together,
these conditions require x2 + y 2 + z 2 < 4. If you
want to wr
2930. These two problems have to do with direction cosine.
Let a be a nonzero vector in R3 . The direction
cosines of a are the three numbers cos , cos , and
cos determined by the angles , , and between
a and, respectively, the positive x-, y-, and z-axes
dicult than using row or column operations to
convert the matrix to a triangular matrix, or various other operations.
I think what Id do is add 7 times the third column to the rst column. That doesnt change the
value of the determinant.
Section 1.6 select
17. Compute the area of the triangle having vertices (1, 0, 1), (0, 2, 3), and (1, 5, 2).
Choose one of the vertices to be the base point,
and let a and b be the displacement vectors from
it to the other two vectors. Then the area of the
1
triangle will b
lie in the plane. The cross product
n = bc
= (1, 1, 3) (2, 1, 2)
= (5, 4, 3)
Section 1.5 selected answers
Math 131 Multivariate Calculus
D Joyce, Spring 2014
is to the plane, so, as in problem 1, a generic
point x in the plane satises the equation
Exercis
way of describing what N is. A complete description couldnt get away with and so forth. If you
want to see all of what and so forth entails, you
can read Dedekinds 1888 paper Was sind und was
sollen die Zahlen? and my comments on it. In that
article he st
Gallery of Surfaces
Math 131 Multivariate Calculus
D Joyce, Spring 2014
There are lots of sufaces well use in this course. Some youre very familiar with like
spheres, cylinders, cones, and tori. (Tori is the plural of torus, torus being Latin.)
Others tha