$\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$. Which of the following statements are true?

I. $\vec{a} \times \vec{b} = -\hat{i} + 5\hat{j} + 3\hat{k}$.

II. $\vec{b} \times \vec{a} = \hat{i} - 5\hat{j} - 3\hat{k}$.

III. The triangle with sides $\vec{a}$ and $\vec{b}$ has area $\frac{\sqrt{35}}{2}$.

IV. $\lvert \vec{a} \times \vec{b} \rvert = \lvert \vec{a} \rvert \lvert \vec{b} \rvert$.

Explanation

1. 01

   $\vec{a} \times \vec{b} = \hat{i}((-1)(-1) - 2(1)) - \hat{j}(1(-1) - 2(2)) + \hat{k}(1(1) - (-1)(2)) = -\hat{i} + 5\hat{j} + 3\hat{k}$: I is true, and II, its negative, is true too.

2. 02

   III: $\frac{1}{2}\sqrt{1 + 25 + 9} = \frac{\sqrt{35}}{2}$, true. IV: $\lvert \vec{a} \rvert \lvert \vec{b} \rvert = \sqrt{6}\sqrt{6} = 6 \ne \sqrt{35}$, because $\vec{a} \cdot \vec{b} = -1 \ne 0$; false.