Page 174 - Advanced engineering mathematics

154 CHAPTER 6 Vectors and Vector Spaces

for t real. These parametric equations are
x =−1 + 8t, y =−1, z = 7 − 3t

for t real. We obtain P 0 when t = 0 and P 1 when t = 1. In this example, the y-coordinate of every
point on the line is −1, so the line is in the plane y =−1.
We may also say that this line consists of all points (−1 + 8t,−1,7 − 3t) for t real.

SECTION 6.1 PROBLEMS

In each of Problems 1 through 5, compute F + G, F − G, 8. 12,(−4,5,1),(6,2,−3)
2F,3G,and F .
9. 4,(0,0,1),(−4,7,5)
√
1. F = 2i − 3j + 5k,G = 2i + 6j − 5k
In each of Problems 10 through 15, find the para-
2. F = i − 3k,G = 4j
metric equations of the line containing the given
3. F = 2i − 5j,G = i + 5j − k points.
√
4. F = 2i − j − 6k,G = 8i + 2k
5. F = i + j + k,G = 2i − 2j + 2k 10. (1,0,4),(2,1,1)
11. (3,0,0),(−3,1,0)
In each of Problems 6 through 9, find a vector having the
given length and in the direction from the first point to the 12. (2,1,1),(2,1,−2)
second. 13. (0,1,3),(0,0,1)
6. 5,(0,1,4),(−5,2,2) 14. (1,0,−4),(−2,−2,5)
7. 9,(1,2,1),(−4,−2,3) 15. (2,−3,6),(−1,6,4)

6.2 The Dot Product

The dot product F · G of F and G is the real number formed by multiplying the two first
components, then the two second components, then the two third components, and adding
these three numbers. If F = a 1 i + b 1 j + c 1 k and G = a 2 i + b 2 j + c 2 k, then
F · G = a 1 a 2 + b 1 b 2 + c 1 c 2 .

Again, this dot product is a number, not a vector. For example,
√ √
( 3i + 4j − πk) · (−2i + 6j + 3k) =−2 3 + 24 − 3π.
The dot product has the following properties.

1. F · G = G · F.
2. (F + G) · H = F · H + G · H.
3. α(F · G) = (αF) · G = F · (αG).
2
4. F · F = F .
5. F · F = 0 if and only if F = O.
2
2
2
2
2
6. αF + βG = α F +2αβF · G + β G .
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October 14, 2010 14:21 THM/NEIL Page-154 27410_06_ch06_p145-186