Cross Products

Calculate the cross product of the following two vectors: U = 2 i + 4 j and V = i + j k

Hint
$$$\vec{a} \times \vec{b}=\begin{bmatrix}a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1\end{bmatrix}$$$
Hint 2
$$$\vec{a}=(a_{1}, a_{2}, a_{3})$$$
$$$\vec{b}=(b_{1}, b_{2}, b_{3})$$$
By finding the cross product, we are calculating the perpendicular vector of U and V, and can use the below formula to get started.
$$$\vec{a} \times \vec{b}=\begin{bmatrix}a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1\end{bmatrix}$$$
where $$\vec{a}=(a_{1}, a_{2}, a_{3})$$ and $$\vec{b}=(b_{1}, b_{2}, b_{3})$$
Thus,
$$$\vec{U} \times \vec{V}=\begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ U_x & U_y & U_z\\ V_x & V_y & V_z\end{bmatrix}=\begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ 2 & 4 & 0\\ 1 & 1 & -1\end{bmatrix}$$$
$$$=\mathbf{i}[(4)(-1)-(0)(1)] - \mathbf{j}[(2)(-1)-(0)(1)] + \mathbf{k}[(2)(1)-(4)(1)]$$$
$$$=\mathbf{i}(-4-0) - \mathbf{j}(-2-0) + \mathbf{k}(2-4)=\{-4;\:2;\:-2 \}$$$
{-4; 2; -2}