Vector Algebra ...

The vector
$$\vec a + \vec b,\vec a - k\vec b$$ where $$k$$ scalar are collinear, for

The position vector of point F, is?

If $$ | \overline{a} | = 1 , | \overline{b} | = 2, | \overline{a} - \overline{b} |^2 + | \overline{a} + 2 \overline{b} |^2 = 20, $$ then $$ ( \overline{a} , \overline{b} ) = $$ 

If $$\bar{a}=2\bar{i}+\bar{j}+\bar{k}, \bar{b}=\bar{i}+5\bar{j}, \bar{c}=4\bar{i}+4\bar{j}-2\bar{k}$$ then the length of the projection of $$(3\bar{a}-2\bar{b})$$ in the direction of $$\bar{c}$$.

In a triangle ABC, if $$ 2\vec { AC } =3\vec { CB }$$, then $$2\vec { OA } +3\vec { OB }$$ equals ?

If $$|\overrightarrow{a}| = 5, |\overrightarrow{a} - \overrightarrow{b}|=8$$ and $$|\overrightarrow{a} + \overrightarrow{b}| = 10$$, then $$|\overrightarrow{b}|$$ is equal to:

If the vector $$OP$$ in $$XY$$ plane whose magnitude is $$\sqrt3$$ makes an angle $$60^o$$ with $$Y-$$ axis, the length of the component of the vector in direction of $$X-$$ axis is :

Let $$\overrightarrow{a}=\hat{i}-\hat{k}, \overrightarrow{b}=x\hat{i}+\hat{j}+\left(1-x\right)\hat{k},\overrightarrow{c}=y\hat{i}+x\hat{j}+\left(1+x-y\right)\hat{k}$$. then $$\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]$$ depends on

For three vectors $$\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$$ which of the following expressions is not equal to any of remaining is

If $$\bar{a}$$ and $$\bar{b}$$ are non-collinear unit vectors and $$\left|\bar{a}+\bar{b}\right|=\sqrt{3}$$ then $$\left(2\bar{a}+5\bar{b}\right)\cdot \left(3\bar{a}-\bar{b}\right)=?$$