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Analytic geometry in space

z (l)

M
− →
− − −→
M 0 M
s
M 0
−→
r

−→ y
r 0 . . . . . . . . . . . .
0

x

Figure 8.11 – Vector equation of a line

−→
Equation (8.19) is called the vector equation of the line. Taking into account that r 0 (x 0 , y 0 ,
−→
z 0 ) and r (x, y, z), let’s write equation (8.19) in the view of coordinates:

x = x 0 + mt,

y = y 0 + nt, (8.20)


z = z 0 + pt.
This equation is called the parametric equation of the line. When changing parameter t the
coordinates of point M(x, y, z) are changing as well, and point M is moving along the line.
Equation (8.20) is used in mechanics, for example when composing the law of movement of the
point along the line. Parameter t in this case acts as time.

8.7. Distance from a point to a line in space

M 1

d
(l)
. . . . . .
−→
M 0 s

Figure 8.12 – Distance from a point to a line in space

Let’s assume we have got line l: x−x 0 = y−y 0 = z−z 0 and point M 1 (x 1 , y 1 , z 1 ). Let’s for-
m n p
mulate the task to obtain the distance from point M 1 to line l. As it is seen from (fig. 8.12), the
− →
sought distance d equals to the height of the parallelogram, built on vectors s = (m, n, p) and
−−−−→
M 0 M 1 = (x 1 − x 0 , y 1 − y 0 , z 1 − z 0 ) As the module of the vector multiplication of these vectors
−→ −→
|M 0 M 1 × s |
equals to the area of the parallelogram, then d = or in the view of coordinates:
−→
| s |
√
2 2 2

y 1 − y 0 x 1 − x 0 z 1 − z 0 x 1 − x 0 x 1 − x 0 y 1 − y 0
+ +
n p p m m n
d = √ (8.21)
2
2
m + n + p 2
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