$\hat i\times\hat j$

Question

Very basic question ahead.

It is required to evaluate the cross product of $\hat i$ and $\hat j$, that is, $\hat i\times\hat j$. Knowing that the cross product is anti-commutative, I made sure to include these in the right order.

Now my work:

\begin{align} \hat i\times\hat j &= (1, 0, 0)\times(0, 1, 0)&&\newline &= \big((0)(0) - (0)(1), (0)(1) - (1)(0), (1)(0) - (1)(1) \big)&&\newline &= (0, 0, -1)&&\newline &= -\hat k \end{align}

But "cheating" and exploiting the right-hand rule clearly shows that the correct answer is $+\hat k$. After trying multiple times, I'm wondering where I have gone wrong in my above process.

Answer 1

My rubric for computation of $\mathbf{x} \times \mathbf{y}$ is $$ \left|\begin{array}z & \mathbf{i}\quad\;\; \mathbf{j}\quad\;\;\mathbf{k} \\ &x_1 \quad x_2 \quad x_3 \\ &y_1 \quad y_2 \quad y_3 \end{array}\right| = \left(x_2 y_3 - y_2 x_3\right)\mathbf{i} +\left(x_3y_1-y_3x_1\right)\mathbf{j} +\left(x_1y_2-y_1x_2\right)\mathbf{k} $$ (The first is the "nonsense" version to help me remember the second one.)
Using this I get $\mathbf{i} \times \mathbf{j}= +\mathbf{k}$ .