DẠNG 3. PHƯƠNG TRÌNH ĐƯỜNG THẲNG

\(\left\{ {\begin{array}{*{20}{l}}{{\rm{x}} = {\rm{a}} + {{\rm{x}}_0}{\rm{t}}}\\{{\rm{y}} = {\rm{b}} + {{\rm{y}}_0}{\rm{t}}.}\\{{\rm{z}} = {\rm{c}} + {{\rm{z}}_0}{\rm{t}}}\end{array}} \right.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = {x_0} + at}\\{y = {y_0} + ct}\\{z = {z_0} + bt}\end{array}} \right..\)

\(\left\{ {\begin{array}{*{20}{l}}{x = {x_0} + at}\\{y = {y_0} + bt}\\{z = {z_0} - ct}\end{array}} \right.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = {x_0} + at}\\{y = {y_0} + bt}\\{z = {z_0} + ct}\end{array}} \right..\)

\(\frac{{x - {x_0}}}{a} = \frac{{y - {y_0}}}{c} = \frac{{z - {z_0}}}{b}.\)

\(\frac{{x - {x_0}}}{a} = \frac{{y - {y_0}}}{b} = \frac{{z - {z_0}}}{c}.\)

\(\frac{{x - {x_0}}}{a} = \frac{{y - {y_0}}}{{ - b}} = \frac{{z - {z_0}}}{c}.\)

\(\frac{{x - {x_0}}}{a} = \frac{{y - {y_0}}}{b} = \frac{{z - {z_0}}}{{ - c}}.\)

\(\cos \varphi = \frac{{\left| {a{a^\prime } + b{b^\prime } + c{c^\prime }} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\sin \varphi = \frac{{a{a^\prime } + b{b^\prime } + {c^\prime }}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {{\rm{c}}^{\prime 2}}} }}.\)

\(\sin \varphi = \frac{{\left| {a{a^\prime } + b{b^\prime } + c{c^\prime }} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\cos \varphi = \frac{{a{a^\prime } + b{b^\prime } + c{c^\prime }}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\cos \alpha = \frac{{\left| {a{a^\prime } + b{b^\prime } + c{c^\prime }} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\sin \alpha = \frac{{a{a^\prime } + b{b^\prime } + c{c^\prime }}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\sin \alpha = \frac{{\left| {a{a^\prime } + b{b^\prime } + c{c^\prime }} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\cos \alpha = \frac{{a{a^\prime } + b{b^\prime } + c{c^\prime }}}{{\sqrt {{a^2} + {b^2} + {c^2}} \cdot \sqrt {{a^{\prime 2}} + {b^{\prime 2}} + {c^{\prime 2}}} }}.\)

\(\overrightarrow {{\rm{BD}}} .\)

\(\overrightarrow {{\rm{AC}}} .\)

\(\overrightarrow {{\rm{AD}}} .\)

\(\overrightarrow {{\rm{CD}}} .\)

\(\overrightarrow {\rm{i}} = (1;0;0).\)

\(\overrightarrow {\rm{j}} = (0;1;0).\)

\(\overrightarrow {\rm{k}} = (0;0;1).\)

\(\overrightarrow {\rm{u}} = (1;1;1).\)

\(\overrightarrow {\rm{i}} = (1;0;0).\)

\(\overrightarrow {\rm{j}} = (0;1;0).\)

\(\overrightarrow {\rm{k}} = (0;0;1).\)

\(\overrightarrow {\rm{u}} = (1;1;1).\)

\(\overrightarrow {\rm{i}} = (1;0;0).\)

\(\overrightarrow {\rm{j}} = (0;1;0).\)

\(\overrightarrow {\rm{k}} = (0;0;1).\)

\(\overrightarrow {\rm{u}} = (1;1;1).\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 3 - t}\\{y = 2 - 2t}\\{z = 3 + t}\end{array}} \right..\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 3 + t}\\{y = 2 - 2t}\\{z = 3 + t}\end{array}} \right..\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 3 + t}\\{y = 2 - 2t}\\{z = 3 - t}\end{array}} \right..\)

\(\left\{ {\begin{array}{*{20}{l}}{x = - 3 + t}\\{y = 2 - 2t}\\{z = 3 + t}\end{array}} \right..\)

\(\frac{{x + 12}}{{17}} = \frac{{y - 13}}{{18}} = \frac{{z + 14}}{{ - 19}}.\)

\(\frac{{x + 12}}{{17}} = \frac{{y - 13}}{{ - 18}} = \frac{{z + 14}}{{19}}.\)

\(\frac{{x + 12}}{{17}} = \frac{{y - 13}}{{18}} = \frac{{z + 14}}{{19}}.\)

\(\frac{{{\rm{x}} + 12}}{{17}} = \frac{{{\rm{y}} - 13}}{{ - 18}} = \frac{{{\rm{z}} + 14}}{{ - 19}}.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 1 - 3t}\\{y = - 1 + t}\\{z = 1 + t}\end{array}} \right.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 1 + 3t}\\{y = - 1 - t}\\{z = 1 + t}\end{array}} \right.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 1 + 3t}\\{y = - 1 - t}\\{z = 1 - t}\end{array}} \right.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 1 + 3t}\\{y = - 1 + t}\\{z = 1 + t}\end{array}} \right..\)

\(\frac{{x + 1}}{1} = \frac{{y - 2}}{6} = \frac{{z - 3}}{8}.\)

\(\frac{{x + 1}}{{ - 1}} = \frac{{y + 2}}{6} = \frac{{z + 3}}{8}.\)

\(\frac{{x - 1}}{1} = \frac{{y + 2}}{{ - 6}} = \frac{{z + 3}}{8}.\)

\(\frac{{x - 1}}{1} = \frac{{y + 2}}{6} = \frac{{z + 3}}{8}.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 1 + 3t}\\{y = - 3 + 2t}\\{z = 2 - t}\end{array}} \right.\)

\(\left\{ {\begin{array}{*{20}{l}}{x = - 1 + 3t}\\{y = 3 + 2t}\\{z = - 2 - t}\end{array}} \right..\)

\(\left\{ {\begin{array}{*{20}{l}}{x = 3 - t}\\{y = 2 + 3t}\\{z = - 1 - 2t}\end{array}} \right..\)

\(\left\{ {\begin{array}{*{20}{l}}{x = - 3 + t}\\{y = - 2 - 3t}\\{z = 1 + 2t}\end{array}} \right..\)

\(\frac{{x - 1}}{1} = \frac{{y + 2}}{2} = \frac{{z + 3}}{{ - 3}}.\)

\(\frac{{x + 1}}{1} = \frac{{y - 2}}{2} = \frac{{z - 3}}{{ - 3}}.\)

\(\frac{{x - 1}}{1} = \frac{{y - 2}}{{ - 2}} = \frac{{z + 3}}{{ - 3}}.\)

\(\frac{{{\rm{x}} - 1}}{1} = \frac{{{\rm{y}} + 2}}{{ - 2}} = \frac{{{\rm{z}} + 3}}{{ - 3}}.\)

\({\rm{d}}:\frac{{{\rm{x}} - 2}}{2} = \frac{{{\rm{y}} - 3}}{{ - 4}} = \frac{{{\rm{z}} - 1}}{{ - 1}}.\)

\({\rm{d}}:\frac{{{\rm{x}} - 2}}{2} = \frac{{{\rm{y}} - 3}}{3} = \frac{{{\rm{z}} - 1}}{1}.\)

\({\rm{d}}:\frac{{{\rm{x}} + 2}}{2} = \frac{{{\rm{y}} + 3}}{{ - 4}} = \frac{{{\rm{z}} + 1}}{{ - 1}}.\)

\({\rm{d}}:\frac{{{\rm{x}} - 2}}{1} = \frac{{{\rm{y}} - 3}}{{ - 1}} = \frac{{{\rm{z}} - 1}}{3}.\)