<span><span /><span><span>
1.
Нахождение длин ребер и координат векторов:
</span><span>
x y
z Длина ребра
</span><span>
Вектор
АВ={xB-xA, yB-yA, zB-zA}
2
-2
-3 4,123105626
</span><span>
Вектор
BC={xC-xB, yC-yB, zC-zB}
2 2 9 9,433981132
</span><span>
Вектор
АC={xC-xA, yC-yA, zC-zA} 4
0 6 7,211102551
</span><span>
Вектор
АD={xD-xA, yD-yA, zD-zA}
5 2
-4 6,708203932
</span><span>
Вектор
BD={xD-xB, yD-yB, zD-zB} 3
4
-1
5,099019514
</span><span>
Вектор
CD={xD-xC, yD-yC, zD-zC}
1 2 -10 10,24695077.
2) АВ*АС = (2*4-2*0-3*6) = -10.
|AB| = </span></span></span>√17, |AC| = √52 = 2√13.
cos (AB∧AC) = -10/(√17*√52) =<span><span />
-10/</span>√<span>884 = -10/29,7321375 </span>≈ -0,336336 AB∧AC = <span><span>
1,9138203 радиан =
</span><span>
109,65382</span><span /></span>°.
3) Проекция вектора АD на вектор АВ.
<span>
Грань АДС. Косинус угла DAC =
<span>-0,0826898.
</span></span>АD1 = АD*cos(<DAC) = 6,708203932*(-0,0826898) = <span><span>-0,554700041.
4) S(ABC) = </span></span><span><span /><span><span>
a1
a2
a3
S
</span><span>ABC
[AB ; AC]=
-12
-24
8 14.
5) </span></span></span><span><span /><span><span><span>Объем пирамиды равен:
</span></span><span><span>(AB{x1, y1, z1} ; AC{x2, y2, z2} ; AD{x3, y3, z3})= x3·a1+y3·a2+z3·a3</span></span></span></span><span> V(ABCD) = </span><span><span /><span><span>
x
y z
</span><span>
AB*AC
-12 -24
8.
</span></span></span>Находим определитель матрицы<span>
∆ =
2*(0*(-4)-2*6)-4*((-2)*(-4)-2*(-3))+5*((-2)*6-0*(-3)) = -140.
</span><span><span> </span><span /><span /><span>V = (1/6) *
140 = 23,3333333.</span><span /></span>