If $|a \times b| = 4$ and $|a \cdot b| = 2$,then $|a|^2 |b|^2 = $

For what value of $x$ is the angle between the vectors $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ acute,and the angle between $\vec{b}$ and the $x$-axis lies between $\pi/2$ and $\pi$?

Let $a, b, c$ be three vectors such that $a \neq 0$,$a \times b = 2a \times c$,$|a| = |c| = 1$,$|b| = 4$,and $|b \times c| = \sqrt{15}$. If $b - 2c = \lambda a$,then $\lambda$ equals:

If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the positive direction of $x, y$ and $z$-axes,then which of the following is a false statement?

If $|a|=2$ and $|b|=3$ and the angle between $a$ and $b$ is $120^{\circ}$,then the length of the vector $\left|\frac{a}{2}-\frac{b}{3}\right|$ is

If $\bar{a} = \hat{i} + \hat{j}$ and $\bar{b} = 2\hat{i} - \hat{k}$,then the point of intersection of the lines $\bar{r} \times \bar{a} = \bar{b} \times \bar{a}$ and $\bar{r} \times \bar{b} = \bar{a} \times \bar{b}$ is