Example 2. The vertices of a pyramid are located at points $O(0,0,0), A(5,2,0), B(2,5,0)$, and $C(1,2,4)$. Calculate its volume, the area of face $A B C$, and the height of the pyramid dropped to this face.
Solution. 1) Let's make a schematic drawing (Fig. 3.15).
2) Introduce vectors $\overrightarrow{A B}=\{-3,3,0\}, \quad \overrightarrow{A C}=\{-4,0,4\}, \quad \overrightarrow{A O}=$ $=\{-5,-2,0\}$. The volume of the pyramid $O A B C$ (tetrahedron) is

$$
V_{T}=\left|\frac{1}{6}\right| \begin{array}{rrr}
-3 & 3 & 0 \\
-4 & 0 & 4 \\
-5 & -2 & 0
\end{array}||=\left|\frac{1}{6}(-84)\right|=14
$$

3) The area of the face $A B C$ :

$$
\left.S_{A B C}=\frac{1}{2}|\overrightarrow{A B} \times \overrightarrow{A C}|=\frac{1}{2}\left|\begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k} \\
-3 & 3 & 0 \\
-4 & 0 & 4
\end{array} \|=\frac{1}{2}\right| 12 \vec{i}+12 \vec{j}+12 \vec{k} \right\rvert\,=6 \sqrt{3}
$$

4) The volume of the pyramid $V_{T}=\frac{1}{3} S_{A B C} \cdot O D$, hence $O D=\frac{3 V_{T}}{S_{A B C}}=$ $=\frac{3 \cdot 14}{6 \cdot \sqrt{3}}=\frac{7 \sqrt{3}}{3}$.

Answer. $\quad 14 ; \quad 6 \sqrt{3} ; \quad \frac{7 \sqrt{3}}{3}$.

## Exercises

1. Show that the points $A(2,-1,-2), B(1,2,1), C(2,3,0)$ and $D(5,0,-6)$ lie in the same plane.

Hint. Check the condition of coplanarity of vectors.

2. Given a pyramid with vertices $A(2,0,0), B(0,3,0), C(0,0,6)$ and $D(2,3,8)$. Calculate its volume and the height dropped to the face $A B C$.
3. Given three consecutive vertices of a parallelogram $A(1,-2,3), B(3,2,1)$ and $C(6,4,4)$. Find its fourth vertex $D$.

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-040.jpg?height=59&width=1064&top_left_y=1208&top_left_x=1411)
coordinates.

4. Determine the angles of $\triangle A B C$ with vertices $A(2,-1,2), B(1,1,1)$ and $C(0,0,5)$.
5. From the vertex of a square, lines are drawn that bisect the opposite sides. Find the angle between these lines.
6. Given vectors $\overrightarrow{O A}$ and $\overrightarrow{O B}$, where $|\overrightarrow{O A}|=2,|\overrightarrow{O B}|=4$, and $\angle A O B=$ $=60^{\circ}$. Determine the angle between the median $\overrightarrow{O M}$ of triangle $A O B$ and the side $\overrightarrow{O A}$.
7. Vectors $\vec{a}$ and $\vec{b}$ form an angle of $45^{\circ}$. Find the area of the triangle constructed on vectors $\vec{a}-2 \vec{b}$ and $3 \vec{a}+2 \vec{b}$, if $|\vec{a}|=|\vec{b}|=5$.
8. Calculate the area of the triangle with vertices $A(7,3,4)$, $B(1,0,6)$ and $C(4,5,-