Question

sejam os vetores u = 3i+2j+6k, v = -4i-2j+k e w = (2u+3v)-(5k). O produto misto [u,v,w] é:

Answer 1

Olá

Resposta correta, letra A) -10

$\vec{u}=\mathsf{3i+2j+6k \qquad \qquad \Longrightarrow \qquad}\boxed{\vec{u}=\mathsf{(3,2,6)}}\\\\\vec{v}=\mathsf{-4i-2j+k \qquad \qquad \Longrightarrow \qquad}\boxed{\vec{v}=\mathsf{(-4,-2,1)}}\\\\\vec{w}=(2\vec{u}+3\vec{v})-5k\\\\2\vec{u}=2\cdot (3,2,6)\qquad\qquad \Longrightarrow 2\vec{u}=(6,4,12)\\\\3\vec{v}=3\cdot (-4,-2,1)\qquad\quad \Longrightarrow 3\vec{u}=(-12,-6,3)\\\\\vec{w}=(6,4,12)+(-12,-6,3)~-~(0,0,5)\\\\\boxed{\vec{w}=(-6,-2,10)}$


Para calcular o produto misto, basta montar uma matriz 3x3 e calcular o determinante.

$[\vec{u},\vec{v},\vec{w}]= \left|\begin{array}{ccc}3&2&6\\-4&-2&1\\-6&-2&10\end{array}\right|\\\\\\ ~[\vec{u},\vec{v},\vec{w}] \mathsf{=\underbrace{(\mathsf{-60-12+48})}_{diag.~principal}~-\underbrace{(\mathsf{-80-6+72})}_{diag.~secund\'aria}}\\\\\\ ~[\vec{u},\vec{v},\vec{w}]=\mathsf{(-72+48)~-~(-86+72)}\\\\\\~[\vec{u},\vec{v},\vec{w}]=\mathsf{-24~-~(-14)}\\\\\\~[\vec{u},\vec{v},\vec{w}]=\mathsf{-24+14}\\\\\\~[\vec{u},\vec{v},\vec{w}]=\mathsf{-10 \qquad\qquad \Longrightarrow \quad\text{Letra A)}}$