Dados los vectores: a=(4;−1;−3);b=2i−3j+2k;c=−3i+2j+7k; determine: c⋅(a×b)+3(c⋅b)−2∥b×c∥
Respuestas a la pregunta
El valor de la operación entre vectores es:
c · (a × b) + 3(c · b) - 2||b × c|| = 46,19
Los vectores:
- a = (4;−1;−3) = 4i-j-3k;
- b = 2i−3j+2k = (2; -3; 2);
- c = −3i+2j+7k = (-3; 2; 7);
Producto vectorial
(a × b) = i j k
4 -1 -3
2 -3 2
(a × b) = i [(-1)(2)-(-3)(-3)] - j [(4)(2)-(2)(-3)] + k [(4)(-3)-(2)(-1)]
(a × b) = i(-11) -j(14) + k(-10)
(a × b) = -11i - 14j - 10k
(b × c) = i j k
2 -3 2
-3 2 7
(b × c) = i [(-3)(7)-(2)(2)] - j [(2)(7)-(-3)(2)] + k [(2)(2)-(-3)(-3)]
(b × c) = -25i + j20 - 5k
||b × c|| = √[(-25)²+(20)²+(5)²]
||b × c|| = 5√42
Producto escalar o punto;
(c · b) = (-3, 2, 7) · (2, -3, 2)
(c · b) = (-3)(2) + (2)(2) + (7)(2)
(c · b) = -6+4+14
(c · b) = 12
c · (a × b) = (-3, 2, 7) · (-11, -14, -10)
c · (a × b) = (-3)(-11) + (2)(-14) + (7)(10)
c · (a × b) = 33-28+70
c · (a × b) = 75
Sustituir;
c · (a × b) + 3(c · b) - 2||b × c|| = 75 + 3(12) -2(5√42)
c · (a × b) + 3(c · b) - 2||b × c|| = 46,19