Question

Dados los vectores: a=(−2;−2;−1);b=2j−3i+2k;c=−i+2j+4k; determine: a⋅(b×c)+2a⋅c−b⋅a

Answer 1

El valor de la operación entre vectores es:

a ⋅ (b × c) + 2a ⋅ c - b ⋅ a  = 21

Los vectores en cuestión son:

• a = (−2;−2;−1)= -2i-2j-k;
• b = 2j−3i+2k = (2; -3; 2);
• c = −i+2j+4k = (-1; 2; 4);

Determine: a⋅(b × c)+2a⋅c−b⋅a

Producto vectorial;

(b × c) =  i     j    k

             2   -3   2

            -1     2   4

(b × c) = i[(-3)(4)-(2)(2)] -j[(2)(4)-(-1)(2)]+k[(2)(2)-(-1)(-3)]

(b × c) = (-16i + 10j + k)

Producto punto;

a⋅(b × c) = (−2;−2;−1)(-16;  10; 1)

a⋅(b × c) = (-2)(-16) + (-2)(10) + (-1)(1)

a⋅(b × c) = 32-20-1

a⋅(b × c) = 11

2a ⋅ c = 2(−2;−2;−1)(-1; 2; 4)

2a ⋅ c = (-4; -4; -2)(-1; 2; 4)

2a ⋅ c = (-4)(-1) + (-4)(2) + (-2)(4)

2a ⋅ c = 4 - 8 - 8

2a ⋅ c = -12

b ⋅ a = (2; -3; 2)(−2;−2;−1)

b ⋅ a =  (2)(-2) + (-3)(-2)  + (2)(-1)

b ⋅ a = -4 + 6 - 24

b ⋅ a = -22

Sustituir;

a ⋅ (b × c) + 2a ⋅ c - b ⋅ a  = 11 + (-12) - (-22)

a ⋅ (b × c) + 2a ⋅ c - b ⋅ a  = 21