Vector Algebra ...

If $$\vec { x } $$ is a vector in the direction of $$(2,-2,1)$$ of magnitude $$6$$ and $$\vec { y } $$ is a vector in the direction of $$(1,1,-1)$$ of magnitude $$\sqrt{3}$$, then $$\left| \vec { x } +2\vec { y }  \right| =...$$

The position vector of a point P is $$\overrightarrow r=x \overrightarrow i + y \overrightarrow j+x \overrightarrow k$$, Where $$x,y,z,\epsilon N$$ and $$\overrightarrow a= \overrightarrow i+ \overrightarrow j+\overrightarrow k$$. If $$\overrightarrow r. \overrightarrow a=10$$, then the number of possible positions of P is ___________.

If the position vector $$\vec{a}$$ of point $$(12, n) $$ is such that $$\left | \vec{a} \right | = 13$$, then find the value (s) of $$n$$.

If a unit vector $$ \vec{a} $$ makes an angle $$ \dfrac{\pi }{3} $$ with $$ \hat{i},\dfrac{\pi }{4} $$ with $$ \hat{j} $$ and an accute angle $$ \theta $$ with $$ \hat{k}, $$ then find $$ \theta $$ and hence, the components of $$ \vec{a} $$.

What is the scalar projection of 
$$\vec{a}=\hat{i}+2\hat{j}+\hat{k}$$ on $$\vec{b}=4\hat{i}+4\hat{j}+7\hat{k}
$$ ?

The adjacent sides of a parallelogram are represented by the vectors $$ \vec{a} = \hat{i}+\hat{j}+\hat{k} $$ and $$ \vec{b} = 2\hat{i}+\hat{j}+2\hat{k}.$$ Find unit vectors parallel to the diagonals of the parallelogram.

Express $$ \vec{AB}$$ in terms of unit vectors $$ \hat{i} $$ and $$\hat{j}$$, when the points are:
A(4,-1), B(1,3)
Find $$ \left | \vec{AB} \right |$$ in each case.

The unit vector normal to the plane containing $$\vec{a}=(\hat{i}-\hat{j}-\hat{k})$$ and $$\vec{b}=(\hat{i}+\hat{j}+\hat{k})$$ is?

If $$\bar{a}$$ and $$\bar{b} = 3 \hat{i} + 6 \hat{j} + 6 \hat{k}$$ are collinear and $$\bar{a} . \bar{b} = 27$$, then $$\bar{a}$$ is equal to 

Let $$O$$ be the circumcentre, $$G$$ be the centroid and $$O$$ be the orthocentre of a $$\triangle ABC$$. Three vectors are taken through $$O$$ and are represented by $$\vec{a}=\vec{OA}, \vec{b}=\vec{OB}$$ and $$\vec{c}=\vec{OC}$$ then $$\vec{a}+\vec{b}+\vec{c}$$ is