Page 172 - Advanced engineering mathematics

152 CHAPTER 6 Vectors and Vector Spaces

P 1
B

y C
x
A u
v
D
P 0
FIGURE 6.8 Quadrilateral
with lines connecting successive FIGURE 6.9 Quadrilateral of
midpoints. Figure 6.8 with vectors as sides.

As an example of the efficiency of vector notation, we will derive a fact about quadrilaterals:
the lines formed by connecting successive midpoints of the sides of a quadrilateral form a paral-
lelogram. Figures 6.8 and 6.9 illustrate what we want to show. Draw the quadrilateral again with
vectors A,B, C, and D as the sides (Figure 6.9). The vectors x,y, u, and v connect the midpoints
of successive sides. We want to show that x and u are parallel and of the same length, and the
same for y and v. From the parallelogram law and the choices of these vectors,
1 1
x = A + B
2 2
and
1 1
u = C + D.
2 2
But also by the parallelogram law, C + D is the vector from P 1 to P 0 , while A + B is the
vector from P 0 to P 1 . These vectors have the same lengths and opposite directions, so
A + B =−(C + D).

Then x =−u, so these vectors are parallel and of the same length (just opposite in direction).
Similarly, y and v are parallel and of the same length.

Equation of a Line in 3-Space
We will show how to find parametric equations of a line L in 3-space containing two given
points. This is more subtle than the corresponding problem in the plane, because there is no
slope to exploit. To illustrate the idea, suppose the points are (−2,−4,7) and (9,1,−7).Form
a vector between these two points (in either order). The arrow from the first to the second point
represents the vector

V = 11i + 5j − 14k.
Because P 0 and P 1 are on L, V is parallel to L, hence to any other vector aligned with L. Now sup-
pose (x, y, z) is any point on L. Then the vector (x + 2)i + (y + 4)j + (z − 7)k from (−2,−4,7)
to (x, y, z) is also parallel to L, hence to V. This vector must therefore be a scalar multiple of V:
(x + 2)i + (y + 4)j + (z − 7)k = tV
= 11ti + 5tj − 14tk

for some scalar t. Since two vectors are equal only when their respective components are equal,
x + 2 = 11t, y + 4 = 5t, z − 7 =−14t.

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October 14, 2010 14:21 THM/NEIL Page-152 27410_06_ch06_p145-186