Points in space: \(A,\) \(B,\) \(C,\) \(D,\) \({P_1},\) \({P_2},\) \({P_3},\) \({P_4}\)

Point coordinates: \(\left( {{x_0},{y_0},{z_0}} \right),\) \(\left( {{x_1},{y_1},{z_1}} \right),\) \(\left( {{x_2},{y_2},{z_2}} \right),\) \(\left( {{x_3},{y_3},{z_3}} \right),\) \(\left( {{x_4},{y_4},{z_4}} \right)\)

Real number: \(\lambda\)

Point coordinates: \(\left( {{x_0},{y_0},{z_0}} \right),\) \(\left( {{x_1},{y_1},{z_1}} \right),\) \(\left( {{x_2},{y_2},{z_2}} \right),\) \(\left( {{x_3},{y_3},{z_3}} \right),\) \(\left( {{x_4},{y_4},{z_4}} \right)\)

Real number: \(\lambda\)

Distance between two points: \(d\)

Area of a triangle: \(S\)

Volume of a pyramid: \(V\)

Area of a triangle: \(S\)

Volume of a pyramid: \(V\)

- A three-dimensional Cartesian coordinate system is formed by a point called the origin (denoted by \(O\)) and a basis consisting of three mutually perpendicular vectors. These vectors define the three coordinate axes: the \(x-,\) \(y-,\) and \(z-\)axis. They are also known as the abscissa, ordinate and applicate axis, respectively. The coordinates of any point in space are determined by three real numbers: \(x\), \(y\), \(z\).
- The distance between two points
\(A\left( {{x_1},{y_1},{z_1}} \right)\) and \(B\left( {{x_2},{y_2},{z_2}} \right)\) in space is determined by the formula

\(d = \left| {AB} \right| =\) \( \Big[ {{{\left( {{x_2} – {x_1}} \right)}^2} }\) \(+\;{ {{\left( {{y_2} – {y_1}} \right)}^2} }\) \(+\;{ {{\left( {{z_2} – {z_1}} \right)}^2}}\Big]^{\large{\frac{1}{2}}\normalsize} \) - Dividing a line segment in the ratio \(\lambda\)

Suppose that the point \(C\left( {{x_0},{y_0},{z_0}} \right)\) divides the segment \(AB\) in the ratio \(\lambda\). The coordinates of the point \(C\) are given by the expressions

\({x_0} = {\large\frac{{{x_1} + \lambda {x_2}}}{{1 + \lambda }}\normalsize},\;\) \({y_0} = {\large\frac{{{y_1} + \lambda {y_2}}}{{1 + \lambda }}\normalsize},\;\) \({z_0} = {\large\frac{{{z_1} + \lambda {z_2}}}{{1 + \lambda }}\normalsize},\) where \(\lambda = {\large\frac{{AC}}{{CB}}\normalsize},\;\) \(\lambda \ne – 1,\)

where \({x_1}\), \({y_1}\), \({z_1}\) are the coordinates of the point \(A\), and \({x_2}\), \({y_2}\), \({z_2}\) are the coordinates of the point \(B\). - The coordinates of the midpoint of the line segment are obtained from the previous formulas at \(\lambda = 1\) and are written as

\({x_0} = {\large\frac{{{x_1} + {x_2}}}{2}\normalsize},\;\) \({y_0} = {\large\frac{{{y_1} + {y_2}}}{2}\normalsize},\;\) \({z_0} = {\large\frac{{{z_1} + {z_2}}}{2}\normalsize},\), where \(\lambda = {\large\frac{{AC}}{{CB}}\normalsize} = 1.\) - Area of a triangle

The area of a triangle with the vertices \({P_1}\left( {{x_1},{y_1},{z_1}} \right),\) \({P_2}\left( {{x_2},{y_2},{z_2}} \right),\) \({P_3}\left( {{x_3},{y_3},{z_3}} \right)\) is found by the formula

\(S =\) \({\large\frac{1}{2}\normalsize}\left[ {{{\left| {\begin{array}{*{20}{c}} {{y_1}} & {{z_1}} & 1\\ {{y_2}} & {{z_2}} & 1\\ {{y_3}} & {{z_3}} & 1 \end{array}} \right|}^2} }\right.+\left.{ {{\left| {\begin{array}{*{20}{c}} {{z_1}} & {{x_1}} & 1\\ {{z_2}} & {{x_2}} & 1\\ {{z_3}} & {{x_3}} & 1 \end{array}} \right|}^2} }\right.+\left.{ {{\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & 1\\ {{x_2}} & {{y_2}} & 1\\ {{x_3}} & {{y_3}} & 1 \end{array}} \right|}^2}}\right] .\) - Volume of a pyramid

The volume of a pyramid whose vertices have the coordinates \({P_1}\left( {{x_1},{y_1},{z_1}} \right)\), \({P_2}\left( {{x_2},{y_2},{z_2}} \right)\), \({P_3}\left( {{x_3},{y_3},{z_3}} \right)\), \({P_4}\left( {{x_4},{y_4},{z_4}} \right)\) is determined by the expression

\(V =\) \(\pm {\large\frac{1}{6}\normalsize}\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & {{z_1}} & 1\\ {{x_2}} & {{y_2}} & {{z_2}} & 1\\ {{x_3}} & {{y_3}} & {{z_3}} & 1\\ {{x_4}} & {{y_4}} & {{z_4}} & 1 \end{array}} \right|\) or \(V = \) \(\pm {\large\frac{1}{6}\normalsize}\left| {\begin{array}{*{20}{c}} {{x_1} – {x_4}} & {{y_1} – {y_4}} & {{z_1} – {z_4}}\\ {{x_2} – {x_4}} & {{y_2} – {y_4}} & {{z_2} – {z_4}}\\ {{x_3} – {x_4}} & {{y_3} – {y_4}} & {{z_3} – {z_4}} \end{array}} \right|.\)

The sign in the right side of the formulas above is chosen so that to get a positive value for the area or volume.