Page 182 - Advanced engineering mathematics

162 CHAPTER 6 Vectors and Vector Spaces

In each of Problems 5 through 9, determine whether the 11. x − y + 2z = 0
points are collinear. If they are not, determine an equation
12. x − 3y + 2z = 9
for the plane containing these points.
13. Let F and G be nonparallel vectors and let R be the
5. (−1,1,6),(2,0,1),(3,0,0) parallelogram formed by representing these vectors as
6. (4,1,1),(−2,−2,3),(6,0,1) arrows from a common point. Show that the area of
this parallelogram is F × G .
7. (1,0,−2),(0,0,0),(5,1,1)
14. Form a parallelepiped (skewed rectangular box) hav-
8. (0,0,2),(−4,1,0),(2,−1,−1)
ing as incident sides the vectors F, G,and H drawn as
9. (−4,2,−6),(1,1,3),(−2,4,5) arrows from a common point. Show that the volume
of this parallelepiped is
In each of Problems 10, 11, and 12, find a vector nor-
mal to the given plane. There are infinitely many such |F · (G × H)|.
vectors.
This quantity is called the scalar triple product of F,
10. 8x − y + z = 12 G,and H.

6.4 The Vector Space R n

For systems involving n variables we may consider n-vectors
< x 1 , x 2 ,··· , x n >

having n components. The jth component of this n-vector is x j and this is a real number. The
n
1
totality of such n-vectors is denoted R and is called “n-space”. R is the real line, consisting
of all real numbers. We can think of numbers as 1-vectors, although we do not usually do this.
3
2
R is the familiar plane, consisting of vectors with two components. And R is in 3-space. R n
has an algebraic structure which will prove useful when we consider matrices, systems of linear
algebraic equations, and systems of linear differential equations.

Two n-vectors are equal exactly when their respective components are equal:
< x 1 , x 2 ,··· , x n >=< y 1 , y 2 ,··· , y n >

if and only if
x 1 = y 1 , x 2 = y 2 ,··· , x n = y n .

Add n-vectors, and multiply them by scalars, in the natural ways:
< x 1 , x 2 ,··· , x n > + < y 1 , y 2 ,··· , y n >=< x 1 + y 1 , x 2 + y 2 ,··· , x n + y n >
and
α< x 1 , x 2 ,··· , x n >=<αx 1 ,αx 2 ,··· ,αx n >.

These operations have the properties we expect of vector addition and multiplication by
n
scalars. If F, G, and H are in R and α an β are real numbers, then
1. F + G = G + F.
2. F + (G + H) = (F + G) + H.
3. F + O = F,

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October 14, 2010 14:21 THM/NEIL Page-162 27410_06_ch06_p145-186