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.
Point direction form of the equation of a line
  1. Two point form of the equation of a line
    \({\large\frac{{x – {x_1}}}{{{x_2} – {x_1}}}\normalsize} = {\large\frac{{y – {y_1}}}{{{y_2} – {y_1}}}\normalsize} =\) \( {\large\frac{{z – {z_1}}}{{{z_2} – {z_1}}}\normalsize}\)
Two point form of the equation of a line
  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.
Equation of a straight line in parametric form
  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.
Angle between two straight 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  \(Aa + Bb + Cc \) \(= 0.\)
Parallel line and plane
  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  \({\large\frac{A}{a}\normalsize} = {\large\frac{B}{b}\normalsize} = {\large\frac{C}{c}\normalsize}.\)
Perpendicular line and plane