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二元一次方程式
- 當
時,
或
就只有一組解。
- 當
時,
或
就有無限多組解。
- 當
時,
或
就無解。
平面向量
平面向量的表示法
- 設兩點坐標
,則![{\displaystyle {\color {Red}{\overrightarrow {AB}}=(x_{2}-x_{1},y_{2}-y_{1})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90c2eb221cd9793542962d03360e3799dd9c3654)
- 兩向量平行:當
,且
時,則![{\displaystyle {\color {Red}{\frac {x_{1}}{x_{2}}}={\frac {y_{1}}{y_{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27832233efc33df4f4a958b57f51ad30a914587f)
- 分點公式:
、
、
- 內分點:
介於
、
之間,
(內分),
- 則
![{\displaystyle {\color {Blue}{\overrightarrow {OP}}={\frac {n}{m+n}}{\overrightarrow {OA}}+{\frac {m}{m+n}}{\overrightarrow {OB}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/388c4c5b70ac9087da1a46150d0142964bd9d311)
- 外分點:
介於
、
之外,
(外分),
- 則
![{\displaystyle {\color {Blue}{\overrightarrow {OP}}={\frac {-n}{m-n}}{\overrightarrow {OA}}+{\frac {m}{m-n}}{\overrightarrow {OB}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a19f0e390fa3a247b47c7578ff978457293986)
- 直線的參數式:過
,向量
平行的直線上點
可表示為
![{\displaystyle {\color {Blue}{\begin{cases}x=x_{0}+at\\y=y_{0}+bt\end{cases}}t\in \mathbb {R} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c59bf67bbb15a4938d3ea0668043f3dd47f026d)
平面向量的內積
- 設
和
,則![{\displaystyle {\color {Blue}{\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cdot \cos \theta }={\color {Red}a_{1}b_{1}+a_{2}b_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f51dff43db824d3f44a0d78327c927bfc7356a)
- 當
,設
,則符合柯西不等式為:![{\displaystyle {\color {Red}(\left[a_{1}\right]^{2}\left[a_{2}\right]^{2})+(\left[b_{1}\right]^{2}\left[b_{2}\right]^{2})\geq (a_{1}b_{1}+a_{1}b_{2})^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a76c4548ffcc6201ad0eca53ec4a9fc69bd342ca)
- 正射影公式:
- 設
對
之正射影
,則![{\displaystyle {\color {Purple}{\vec {c}}={\Bigg (}{\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|^{2}}}{\Bigg )}{\vec {b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67740ebe6930d7b284a026e3fff8979dfc2953b9)
- 設
對
之正射影
,則![{\displaystyle {\color {Purple}{\vec {c}}={\Bigg (}{\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {a}}\right|^{2}}}{\Bigg )}{\vec {a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3a0f436ca3ee468d81f78aa5ceffa4602d1f98)
- 距離公式:
- 設點
到直線
的距離為 ![{\displaystyle d(P,L)={\frac {\left|ax_{0}+by_{0}+c\right|}{\sqrt {a^{2}+b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c451a5db55639b3ddac3d5cb3fb8226d900c88d4)
- 設直線
的距離為 ![{\displaystyle d(L_{1},L_{2})={\frac {\left|c_{1}c_{2}\right|}{\sqrt {a^{2}+b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1c6fd7154eb833d9eeaacd55086727c5ebb68b)
二階行列式
- 公式:
![{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f341e2a592db51542cfcfb6def112377a86986ae)
- 當解析失败 (未知函数“\begin{cases}”): {\displaystyle \begin{cases} {\color{Red}a_1}x+{\color{Blue}b_1}y={\color{Olive Green}c_1} \\ {\color{Orange Red}a_2}x+{\color{Navy Blue}b_2}y={\color{Emerald}c_2} \end{cases} }
時,
- ∵解析失败 (未知函数“\begin{vmatrix}”): {\displaystyle \vartriangle= \begin{vmatrix} {\color{Red}a_1} & {\color{Blue}b_1} \\ {\color{Orange Red}a_2} & {\color{Navy Blue}b_2} \end{vmatrix} ={\color{Red}a_1} \cdot {\color{Navy Blue}b_2}-{\color{Orange Red}a_2} \cdot {\color{Blue}b_1}}
- 解析失败 (未知函数“\begin{vmatrix}”): {\displaystyle \vartriangle_x= \begin{vmatrix} {\color{Olive Green}c_1} & {\color{Blue}b_1} \\ {\color{Emerald}c_2} & {\color{Navy Blue}b_2} \end{vmatrix} ={\color{Olive Green}c_1} \cdot {\color{Navy Blue}b_2}-{\color{Emerald}c_2} \cdot {\color{Blue}b_1}}
- 解析失败 (未知函数“\begin{vmatrix}”): {\displaystyle \vartriangle_y= \begin{vmatrix} {\color{Red}a_1} & {\color{Olive Green}c_1} \\ {\color{Orange Red}a_2} & {\color{Emerald}c_2} \end{vmatrix} ={\color{Red}a_1} \cdot {\color{Emerald}c_2}-{\color{Orange Red}a_2} \cdot {\color{Olive Green}c_1}}
- ∴
![{\displaystyle x={\frac {\vartriangle _{x}}{\vartriangle }},y={\frac {\vartriangle _{y}}{\vartriangle }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d880e5d65412e4da2e02387afb0aeb405ba02abd)