M_0(x_0,y_0,z_0)

M_1(2,3,2),M_2(3,3,3),M_3(4,5,5)

M1​(2,3,2),M2​(3,3,3),M3​(4,5,5)

\vec{n}

\frac{x}{a}\frac{y}{b}\frac{z}{c}1

ax​by​cz​1

cos\theta\frac{|\vec{n_1}.\vec{n_2}|}{|\vec{n_1}|.|\vec{n_2}|}

cosθ∣n1​

cos\theta\frac{|A_1A_2B_1B_2C_1C_2|}{|\sqrt{A_1^2B_1^2C_1^2}.\sqrt{A_2^2B_2^2C_2^2}|}

cosθ∣A12​B12​C12​

\frac{A_1}{A_2}\frac{B_1}{B_2}\frac{C_1}{C_2}

已知1平面

\frac{3}{\sqrt{6}.\sqrt{6}}\frac{1}{2}

cosθ6

M0​x0​,y0​,z0​是平面AxByCzD0外的一点则点

M_0

d\frac{|Ax_0By_0Cz_0D|}{\sqrt{A^2B^2C^2}}

dA2B2C2

d\frac{|\vec{M_1M_0}.\vec{n}|}{\vec{n}}

∣M1​M0​

\frac{|A(x_0-x_1)B(y_0-y_1)C(z_0-z_1)|}{\sqrt{A^2B^2C^2}}

\downarrow

​∣A(x0​−x1​)B(y0​−y1​)C(z0​−z1​)∣​↓其中Ax1​By1​Cz1​D0

d\frac{|Ax_0By_0Cz_0D|}{\sqrt{A^2B^2C^2}}

dA2B2C2

\frac{|2-43-3|}{\sqrt{2^2(-2)^21^2}}

\frac{2}{3}

A_1xB_1yC_1zD_10,A_2xB_2yC_2zD_20

A1​xB1​yC1​zD1​0,A2​xB2​yC2​zD2​0则方程组

\begin{cases}

{A1​xB1​yC1​zD1​0A2​xB2​yC2​zD2​0​

\frac{x-x_0}{m}\frac{y-y_0}{n}\frac{z-z_0}{p}

mn0,p

\frac{x-3}{1}\frac{y2}{0}\frac{z-6}{0}

1x−3​0y2​0z−6​

\frac{x-x_1}{x_2-x_1}\frac{y-y_1}{y_2-y_1}\frac{z-z_1}{z_2-z_1}

x2​−x1​x−x1​​y2​−y1​y−y1​​z2​−z1​z−z1​​

称方程组为直线

\frac{x-x_0}{m}\frac{y-y_0}{n}\frac{z-z_0}{p}t

\begin{cases}

\frac{x-1}{4}\frac{y}{-1}\frac{z2}{-3}t

4x−1​−1y​−3z2​t

cos\theta\frac{|\vec{s_1}.\vec{s_2}|}{|\vec{s_1}|.|\vec{s_2}|}

cosθ∣s1​

cos\theta\frac{|m_1m_2n_1n_2z_1z_2|}{\sqrt{m_1^2n_1^2p_1^2}\sqrt{m_2^2n_2^2p_2^2}}

cosθm12​n12​p12​

L_1//L_2--\vec{s}//\vec{s_2}--\frac{m_1}{m_2}\frac{n_1}{n_2}\frac{p_1}{p_2}

L1​//L2​−−s

\vec{n_1}(1,-3,0),\vec{n_2}(3,1,6)

n1​

​(3,1,6),因为L1//L3,则可取所求直线的一个方向向量

\left[

\frac{x-3}{-9}\frac{y2}{-3}\frac{z-6}{5}

−9x−3​−3y2​5z−6​

\frac{x-1}{1}\frac{y}{-4}\frac{z-6}{1}

L1​:1x−1​−4y​1z−6​

cos\vartheta\frac{\vec{s_1}\vec{s_2}}{|\vec{s_1}||\vec{s_2}|}\frac{\sqrt{2}}{2}

cosϑ∣s1​

sin\theta|cos(\vec{n},\vec{s})|\frac{|\vec{n}.\vec{s}|}{|\vec{n}||\vec{s}|}

sinθ∣cos(n

sin\theta\frac{AmBnCp}{\sqrt{A^2B^2C^2}\sqrt{m^2n^2p^2}}

sinθA2B2C2

\vec{s}//\vec{n}--\vec{s}\vec{n}0--\frac{A}{m}\frac{B}{n}\frac{C}{p}

L⊥∏−−s

\frac{x-1}{2}\frac{y2}{-3}\frac{z-4}{1}

2x−1​−3y2​1z−4​

\vec{s}//\vec{n}--\vec{s}\vec{n}0--\frac{A}{m}\frac{B}{n}\frac{C}{p}

L⊥∏−−s

[外链图片转存中…(img-ITGnnj8A-1694230678106)]

2x-3yz-40

\frac{x-1}{2}\frac{y2}{-3}\frac{z-4}{1}

2x−1​−3y2​1z−4​