Question

Calcule o produto vetorial u × v para: u = u = (1, −1, 1) e v = (2, −3, 4);

Answer 1

O produto vetorial é dado pelo seguinte determinante:

$\bold{u} \times \bold{v} = \left|\begin{array}{ccc}\bold{\hat{i}}&\bold{\hat{j}}&\bold{\hat{k}}\\u_i&u_j&u_k\\v_i&v_j&v_k\end{array}\right|$

Substituindo os valores, temos:

$\bold{u} \times \bold{v} = \left|\begin{array}{ccc}\bold{\hat{i}}&\bold{\hat{j}}&\bold{\hat{k}}\\1&-1&1\\2&-3&4\end{array}\right| = [-4\bold{\hat{i}} + 2\bold{\hat{j}} - 3\bold{\hat{k}}] - [-3\bold{\hat{i}} + 4\bold{\hat{j}} - 2\bold{\hat{k}}]\\\\\bold{u} \times \bold{v} = -4\bold{\hat{i}} + 2\bold{\hat{j}} - 3\bold{\hat{k}} + 3\bold{\hat{i}} - 4\bold{\hat{j}} + 2\bold{\hat{k}} = -\bold{\hat{i}} - 2\bold{\hat{j}} - \bold{\hat{k}}\\\\\bold{u} \times \bold{v} = (-1, -2, -1)$