Question

Considere os vetores U = (2,a,−1), V = (3,1,−2) e W = (2a − 1,−2,4). Determine a de modo que U.V = (U + V ).(V + W).

Answer 1

Considerando os vetores U, V e W, queremos que:

$U\cdot V = (U + V)\cdot(V + W)$

Então:

$(2,a,-1)\!\cdot\! (3,1,-2)\! =\! ((2,a,-1)\! +\! (3,1,-2))\!\cdot\!((3,1,-2)\!+\!(2a-1,-2,4))\\\\ (2,a,-1)\!\cdot\! (3,1,-2)\! = \! (5,a+1,-3)\!\cdot\!(2a+2,-1,2)\\\\ 2\cdot3+a\cdot1+(-1)\cdot(-2)= 5\cdot(2a+2)+(a+1)\cdot(-1)+(-3)\cdot2\\\\ 6+a+2= 10a+10-a-1-6\\\\ 8a=5\Longrightarrow \boxed{a=\dfrac{5}{8}}$

Portanto, o valor de a para atender à exigência do enunciado é 5/8.