Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \times \vec{a}) = \alpha \hat{i} - \beta \hat{j} + \gamma \hat{k}$, then what is the value of $\alpha + \beta + \gamma$?

If a vector of magnitude 2 units makes an angle $\frac{\pi}{3}$ with $2\hat{i}$, $\frac{\pi}{4}$ with $3\hat{j}$ and an acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector?

Consider the following in respect of moment of a force:

1. The moment of force about a point is independent of point of application of force.

2. The moment of a force about a line is a vector quantity.

Which of the statements given above is/are correct?

For any vector $\vec{r}$, what is $\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdot\hat{j}\right)\left(\vec{r}\times\hat{j}\right) + \left(\vec{r}\cdot\hat{k}\right)\left(\vec{r}\times\hat{k}\right)$ equal to?