If $\overrightarrow{ P }=3 \hat{ i }+\sqrt{3} \hat{ j }+2 \hat{ k }$ and $\overrightarrow{ Q }=4 \hat{ i }+\sqrt{3} \hat{ j }+2.5 \hat{ k }$ then, The unit vector in the direction of $\overrightarrow{ P } \times \overrightarrow{ Q }$ is $\frac{1}{x}(\sqrt{3} \hat{i}+\hat{j}-2 \sqrt{3} \hat{k})$. The value of $x$ is

Show that the area of the triangle contained between the vectors $a$ and $b$ is one half of the magnitude of $a \times b .$

The angle made by the vector $\left( {\hat i\,\, + \;\,\hat j} \right)$ with $x-$ axis and $y$ axis is

Write the distributive law for the product of two vectors. 

If a vector $2\hat i + 3\hat j + 8\hat k$ is perpendicular to the vector $4\hat j - 4\hat i + \alpha \hat k$. Then the value of $\alpha $ is

Vectors $a \hat{i}+b \hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+4 \hat{k}$ are perpendicular to each other when $3 a+2 b=7$, the ratio of a to $b$ is $\frac{x}{2}$. The value of $x$ is $..............$