2.4 Spatial and linear equation
First, the linear equation of the form
\ ((1) \) point and determining a nonzero vector
\(∀M(x,y,z),M_0(x_0,y_0,z_0)∈R^3\)
\(M∈L\iff\vec{M_0M}\parallel\vec s\iff\vec{M_0M}=t\vec s\)
Vector equation \ ((the Vector \; Equation) \) : \ (\ R & lt VEC = \ VEC R_0} + {T \ VEC S \) (point to a formula)
令 \(\vec s=(m,n,p)\Longrightarrow{x-x_0,y-y_0,z-z_0}=t{m,n,p}\)
Parametric equation \ ((from Parametric \; Equation) \) : \ (\ Cases the begin {X} = Y = x_0 y_0 \\ + + (TM) TN \; \; \; \; (- \ infty <T <+ \ infty ) \\ z = z_0 + tp \ end {cases} \)
Wherein \ (T \) as a parameter
By the \ ((x-x_0, y -y_0, z-z_0) \ parallel (m, n, p) \) to give
Symmetrical equation \ ((Symmetrical \; Equation) \) : \ (\ FRAC {X-x_0} {m} = \ FRAC {Y-y_0} {n-} = \ FRAC {Z-Z_0} {P} \)
Standard equation of the line is generally worthwhile to point symmetric equation or equations formula
\ ((2) \) two non-parallel intersecting plane determined
\(\prod_1:A_1x+B_1+C_1z+D_1=0\)
\(\prod_2:A_2x+B_2+C_2z+D_2=0\)
\ (\ Begin {n_1} = (A_1, B_1, C_1) \ nparallel \ begin {N_2} = (A_2, B_2, C_2) \)
General equation: \ (L = \ {Cases} A_1x the begin B_1 + + + C_1z D_1 B_2 = 0 \\ A_2x + + + C_2z D_2 = 0 \ Cases End {} \)
Wherein \ (L \) direction vector \ (\ vec s = \ vec {n_1} \ times \ vec {n_2} = (A_1, B_1, C_1) \ times (A_2, B_2, C_2) \)
\ (L \) is symmetrical: \ (\ FRAC {X-x_0} {B_1C_2-B_2C_1} = \ FRAC {Y-y_0} {C_1A_2-C_2A_1} = \ FRAC {X-x_0} {A_1B_2-A_2B_1} \ )
Further may be derived, two-point equation: \ (\ X FRAC-x_1 {} {} x_1 = x_2-\ FRAC-Y_1 {Y}} {Y_2-Y_1 = \ Z-Z_1 FRAC {} {} Z_2-Z_1 \ )