Vector Algebra ...

Let $$\vec {a} = x\hat {i} + 12\hat {j} - \hat {k}, \vec {b} = 2\hat {i} + 2x\hat {j} + \hat {k}$$ and $$\vec {c} = \hat {i} + \hat {k}$$. If ordered set $$[\vec {b} \vec {c} \vec {a}]$$ is left handed, then.

Which is a unit vector?

If $$\vec {a}$$ and $$\vec {b}$$ are unit vectors, then angle between $$\vec {a}$$ and $$\vec {b}$$ for $$\sqrt {3} \vec {a} - \vec {b}$$ to be unit vector is

If $$ \vec a+2\vec b+3 \vec c =0$$, then $$\vec{b}\times \vec{c}+\vec{c}\times \vec{a}+\vec{a} \times \vec{b}$$ equals to:

Let $$\overline a ,\overline b ,\overline c $$ be vectors of length $$3, 4, 5 $$respectively. Let $$\overline a $$ be perpendicular to $$\overline b  + \overline c ,\overline b \,to\,\overline c  + \overline a \,{\text{and}}\,\overline c \,{\text{to}}\,\overline a  + \overline b .\,{\text{Then}}|\overline a  + \overline b  + \overline c |$$ is equals to:

If the projection of $$\vec{a}$$ on $$\vec{b}$$ and the projection of $$\vec{b}$$ on $$\vec{a}$$ are equal then the angle between $$\vec{a}+\vec{b}$$ and $$\vec{a}-\vec{b}$$ is

If $$\vec a$$ is perpendicular to $$\vec b$$ and $$\vec c,$$ then

Let $$\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \,\,\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$ and $$\vec{c} = \vec{i} + \vec{j} - 2\vec{k} $$ be three vectors. A vector of the type $$\vec{b} + \lambda \vec{c}$$ for some scalar $$\lambda$$, whose projection on $$\vec{a}$$ is of magnitude $$\sqrt {\frac{2}{3}}$$. Thenthe value of $$\lambda$$ is

the resultant of two vectors $$\overline {A\,} {\text{and}}\,\overline B $$ makes an angle $$\alpha \,{\text{with}}\,\overline A $$ and $$\beta \,with\,\overline B $$. If $$A = |\overline A |,B = |\overline B |$$, then 

If $$\overrightarrow a =3\widehat{i}-2\widehat{j}+\widehat{k}$$ and $$\overrightarrow b =2\widehat{i}-4\widehat{j}-3\widehat{k}$$, find $$|\overrightarrow a -2\overrightarrow b|$$.