Question

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∥

Answer 1

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