Vector Algebra ...

If $$\bar{a}$$ is unit vector, then $$|\bar{a}\times \hat{i}|^2+|\bar{a}\times \hat{j}|^2+|\bar{a}\times \hat{k}|^2=$$ _____________.

If $$\vec { a } $$ and $$\vec { b } $$ are non-zero non-collinear vectors, then $$\left[ \vec { a } \quad \vec { b } \quad \hat { i }  \right] \hat { i } +\left[ \vec { a } \quad \vec { b } \quad \hat { j }  \right] \hat { j } +\left[ \vec { a } \quad \vec { b } \quad \hat { k }  \right] \hat { k } $$ is equal to

The vector $$z = 3 - 4i$$ is turned anticlockwise through an angle of $$180^{\circ}$$ and stretched $$\dfrac{5}{2}$$ times. The complex number corresponding to the newly obtained vector is ....

If A and B are the points $$(2,1,-2),(3,-4,5)$$, then the angle that $$OA$$ makes with $$OB$$ is:

If $$\vec {a}.\vec {b}.\vec {c}$$ represents the vectors $$\vec {BC}.\vec {CA}.\vec {AB}$$ respectively, then which one is correct 

If $$\left| {\widehat a - \widehat b} \right| = \sqrt 3 $$ , then  $$\left| {\widehat a + \widehat b} \right|$$  may be:-

A line passes through the points whose position vectors $$ \hat { i } +\hat { j } -2\hat { k }$$ and $$\hat { i } -3\hat { j } +\hat { k }$$. Then the position vector of a point on it at a unit distance from the first point is 

If $$\vec{a}+\vec{b}\perp \vec{a}$$ and $$|\vec{b}|=\sqrt{2}|\vec{a}|$$, then?

Let $$a=\hat{i}+2\hat j+3\hat k$$ and $$b=3\hat i+\hat j$$. Find the unit vector in the direction of the $$a+b$$.

The set of values of $$c$$ for which the angle between the vectors $$cx\hat{i}-6\hat{j}+3\hat{k}$$ and $$x\hat{i}-2\hat{j}+2cx\hat{k}$$ is acute for every $$x\in R$$ is