AB, ,AC AD vektorlarni aralash ko’paytmasini va ABCD piramidani hajmini. 1. A(7;-4;-3), B(-1;8;-1), C(-12;-1, 0), D(2;1, 4)

Vektorlarni topish uchun, A, B, C va D nuqtalari orasidagi vektorlarni topamiz:

\(\overrightarrow{AB} = B - A = (-1 - 7, 8 - (-4), -1 - (-3)) = (-8, 12, 2)\)

\(\overrightarrow{AC} = C - A = (-12 - 7, -1 - (-4), 0 - (-3)) = (-19, 3, 3)\)

\(\overrightarrow{AD} = D - A = (2 - 7, 1 - (-4), 4 - (-3)) = (-5, 5, 7)\)

Vektorlarni aralash ko'paytmasini topish uchun, ularni ko'paytiramiz:

\(\overrightarrow{AB} \cdot \overrightarrow{AC} = (-8)(-19) + (12)(3) + (2)(3) = 152 + 36 + 6 = 194\)

Piramidaning hajmini topish uchun, ABC to'g'ri to'rtburchakning hajmini topish formulani foydalanamiz. ABCD to'g'ri to'rtburchak hajmini topish formulasi:

\[ V = \frac{1}{3} \cdot \left| \overrightarrow{AB} \cdot \overrightarrow{AC} \times \overrightarrow{AD} \right| \]

\[ V = \frac{1}{3} \cdot \left| \begin{vmatrix} i & j & k \\ -8 & 12 & 2 \\ -19 & 3 & 3 \end{vmatrix} \right| \]

\[ V = \frac{1}{3} \cdot \left| (12 \cdot 3 - 2 \cdot 3)i - (-8 \cdot 3 - 2 \cdot -19)j + (-8 \cdot 3 - 12 \cdot -19)k \right| \]

\[ V = \frac{1}{3} \cdot \left| 30i + 22j - 204k \right| \]

\[ V = \frac{1}{3} \cdot \sqrt{30^2 + 22^2 + (-204)^2} \]

\[ V = \frac{1}{3} \cdot \sqrt{900 + 484 + 41616} \]

\[ V = \frac{1}{3} \cdot \sqrt{43000} \]

\[ V = \frac{\sqrt{43000}}{3} \]

\[ V = \frac{10\sqrt{43}}{3} \]

Javob: Piramidaning hajmi \(\frac{10\sqrt{43}}{3}\) ga teng.