Diketahui persamaan vektor:

$A = B \times C$

dengan:

$A = 4i - 2j + 12k \quad \text{dan} \quad C = 4i + 8j + 6k.$

Kita perlu mencari vektor $B$ menggunakan persamaan cross product $A = B \times C$. Untuk itu, kita dapat menggunakan rumus umum perkalian silang dua vektor dalam bentuk determinan:

$B \times C = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}$

Vektor $C$ memiliki komponen $C_x = 4$, $C_y = 8$, dan $C_z = 6$. Vektor $A$ adalah hasil dari perkalian silang, sehingga kita harus menyelesaikan sistem persamaan untuk menemukan komponen $B_x$, $B_y$, dan $B_z$.

Mari kita selesaikan langkah demi langkah dengan menghitung komponen vektor $B$.There was an error when attempting to calculate the cross product. I'll correct this by recalculating the cross product manually and solving for the components of $B$.

The cross product $B \times C$ can be computed using the determinant of the matrix formed by the unit vectors and the components of $B$ and $C$. I'll now proceed with that correction.It appears that the system of equations doesn't yield a unique solution, which suggests that the vectors $A$ and $C$ may not be fully independent, and no unique solution exists for vector $B$ based on the provided information. This can occur in certain cases, such as when the vectors $A$ and $C$ are parallel or when the cross product yields a result that doesn't fit the given conditions.

If you'd like, I can explore alternative interpretations or recheck the conditions. Do you have any further details, or should we proceed with more general questions on this topic?