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Analytic geometry in space

Let’s assume we have got plane (p), given by its general equation Ax + By + Cz + D = 0,
and line l, given by its canonical equation

x − x 0 y − y 0 z − z 0
= = .
m n p

Angle φ between line l and the plane (p) is considered as the sharp angle between l and it’s
−→
projection on plane (p) (fig. 8.13). Let’s write angle θ between vectors n = (A; B; C) and
−→ −→
−→ | n · s | ◦
s = (m; n; p), which is obtained according to the formula: cos θ = . If θ < 90 , then
−→
−→
| n |·| s |
◦
◦
◦
cos θ = cos(90 − φ) = sin φ. If θ > 90 , then cos θ = cos(90 + φ) = − sin φ. In any case
−→ −→
| n · s |
sin φ = |cos θ| . Thus, sin φ = or in the view of coordinates:
−→
−→
| n |·| s |
|Am + Bn + Cp|
sin φ = √ √ (8.25)
2 2 2 2 2 2
A + B + C · m + n + p
is a formula for evaluating of angle between a line and a plane.
8.9.2. Conditions of parallelity and perpendicularity of a line and a plane
Suppose we have line l x−x 0 = y−y 0 = z−z 0 and plane (p) Ax + By + Cz + D = 0. They are
m n p
−→
parallel between each other when and only when directed vector s = (m, n, p) is perpendicular
−→
to normal vector n = (A; B; C) (fig. 8.14). According to the condition of perpendicularity of
−→ −→
vectors: n · s = 0 or in the view of coordinates:
Am + Bn + Cp = 0. (8.26)

− →
s
z

−→ (l)
n

(p)

y
. . . . . . . . .
0

x

Figure 8.14 – Parallelity of a line and a plane

This is the condition of parallelity of the line and the plane.
−→
Line l and plane (p) are perpendicular between each other when and only when vectors s
−→
and n are colinear (fig. 8.15). According to the condition of colinearity:
A B C
= = . (8.27)
m n p

This is the condition of perpendicularity of the line and the plane.

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