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# Formulas and Tables

Analytic Geometry

# Straight Line in Space

Point coordinates: $$x,$$ $$y,$$ $$z,$$ $${x_1},$$ $${y_1},$$ $${z_1},$$ $$\ldots$$
Real numbers: $$A,$$ $$B,$$ $$C,$$ $$D,$$ $${A_1},$$ $${B_1},$$ $$t,$$ $$a,$$ $$b,$$ $$c,$$ $${a_1},$$ $${b_1},$$ $$\ldots$$
Direction vectors of straight lines: $$\mathbf{s},$$ $$\mathbf{s_1},$$ $$\mathbf{s_2}$$

Direction cosines: $$\cos\alpha,$$ $$\cos\beta,$$ $$\cos\gamma$$
Normal vector to a plane: $$\mathbf{n}$$
Angle between two lines: $$\varphi$$

1. Point direction form of the equation of a line
$${\large\frac{{x – {x_1}}}{a}\normalsize} = {\large\frac{{y – {y_1}}}{b}\normalsize} =$$ $${\large\frac{{z – {z_1}}}{c}\normalsize},$$
where the point $${P_1}\left( {{x_1},{y_1},{z_1}} \right)$$ lies on the straight line and the vector $$\mathbf{s}\left( {a,b,c} \right)$$ is direction vector of the line.
1. Two point form of the equation of a line
$${\large\frac{{x – {x_1}}}{{{x_2} – {x_1}}}\normalsize} = {\large\frac{{y – {y_1}}}{{{y_2} – {y_1}}}\normalsize} =$$ $${\large\frac{{z – {z_1}}}{{{z_2} – {z_1}}}\normalsize}$$
1. Equation of a straight line in parametric form
\left\{ \begin{aligned} x &= {x_1} + t\cos\alpha \\ y &= {y_1} + t\cos\beta \\ z &= {z_1} + t\cos\gamma \end{aligned} \right.,
where the point $${P_1}\left( {{x_1},{y_1},{z_1}} \right)$$ lies on the line and $$\cos\alpha,$$ $$\cos\beta,$$ $$\cos\gamma$$ are the direction cosines of the direction vector of the line, the parameter $$t$$ is any real number.
1. Angle between two straight lines
$$\cos \varphi = {\large\frac{{{\mathbf{s_1}} \cdot {\mathbf{s_2}}}}{{\left| {{\mathbf{s_1}}} \right| \cdot \left| {{\mathbf{s_2}}} \right|}}\normalsize} =$$ $${\large\frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \cdot \sqrt {a_2^2 + b_2^2 + c_2^2} }}\normalsize},$$
where $${\mathbf{s_1}}\left( {{a_1},{b_1},{c_1}} \right)$$, $${\mathbf{s_2}}\left( {{a_2},{b_2},{c_2}} \right)$$ are the direction vectors of the lines.
1. Parallel lines
Two straight lines are parallel if their direction vectors $${\mathbf{s_1}}\left( {{a_1},{b_1},{c_1}} \right)$$ and $${\mathbf{s_2}}\left( {{a_2},{b_2},{c_2}} \right)$$ are collinear:
$${\mathbf{s_1}}\parallel {\mathbf{s_2}}$$  or  $${\large\frac{{{a_1}}}{{{a_2}}}\normalsize} = {\large\frac{{{b_1}}}{{{b_2}}}\normalsize} = {\large\frac{{{c_1}}}{{{c_2}}}\normalsize}.$$
2. Perpendicular lines
Two straight lines are perpendicular if the dot product of their direction vectors $${\mathbf{s_1}}\left( {{a_1},{b_1},{c_1}} \right)$$ and $${\mathbf{s_2}}\left( {{a_2},{b_2},{c_2}} \right)$$ is equal to zero:
$${\mathbf{s_1}} \cdot {\mathbf{s_2}} = 0$$  or  $${a_1}{a_2} = {b_1}{b_2} =$$ $${c_1}{c_2}$$ $$= 0.$$
3. Intersection of two lines in space
Two straight lines
$${\large\frac{{x – {x_1}}}{{{a_1}}}\normalsize} = {\large\frac{{y – {y_1}}}{{{b_1}}}\normalsize} = {\large\frac{{z – {z_1}}}{{{c_1}}}\normalsize}$$ and $${\large\frac{{x – {x_2}}}{{{a_2}}}\normalsize} = {\large\frac{{y – {y_2}}}{{{b_2}}}\normalsize} = {\large\frac{{z – {z_2}}}{{{c_2}}}\normalsize}$$
intersect if the following condition is satisfied:
$$\left| {\begin{array}{*{20}{c}} {{x_2} – {x_1}} & {{y_2} – {y_1}} & {{z_2} – {z_1}}\\ {{a_1}} & {{b_1}} & {{c_1}}\\ {{a_2}} & {{b_2}} & {{c_2}} \end{array}} \right|$$ $$= 0.$$
4. Parallel line and plane
The straight line and plane given respectively by the equations
$${\large\frac{{x – {x_1}}}{{a}}\normalsize} = {\large\frac{{y – {y_1}}}{{b}}\normalsize} = {\large\frac{{z – {z_1}}}{{c}}\normalsize}$$ and $$Ax + By + Cz$$ $$+\, D$$ $$= 0,$$
are parallel if
$$\mathbf{n} \cdot \mathbf{s} = 0$$  or  $$Aa + Bb + Cc$$ $$= 0.$$
1. Perpendicular line and plane
The straight line and plane given respectively by the equations
$${\large\frac{{x – {x_1}}}{{a}}\normalsize} = {\large\frac{{y – {y_1}}}{{b}}\normalsize} = {\large\frac{{z – {z_1}}}{{c}}\normalsize}$$ and $$Ax + By + Cz$$ $$+\, D$$ $$= 0,$$
are perpendicular if
$$\mathbf{n}\parallel \mathbf{s}$$  or  $${\large\frac{A}{a}\normalsize} = {\large\frac{B}{b}\normalsize} = {\large\frac{C}{c}\normalsize}.$$