Vector Algebra ...

The vector $$(\hat {i}\times \vec {a}.\vec {b})\hat {i} + (\hat {j} \times \vec {a}.\vec {b})\hat {j} + (\hat {k} \times \vec {a} . \vec {b})\hat {k}$$ is equal to

The x-y plane divides the line joining the points $$(-1, 3, 4)$$ and $$(2, -5, 6)$$:

If $$\vec {a},\vec {b}$$ and $$\vec {c}$$ are those mutually perpendicular vectors, then the projection of the vector $$\left( l \dfrac{\bar {a}}{|\bar {a}|}+m\dfrac{\bar {b}}{|\bar {b}|}+n\dfrac{(\bar {a}\times \bar {b})}{|\bar {a} \times \bar {b}|}\right)$$ along bisector of vectors $$\vec {a}$$ and $$\vec {a}$$ may be given as  ?

$$\bar{a}, \bar{b}, \bar{c}$$ are mutually perpendicular unit vectors and $$\bar{d}$$ is a unit vector equally inclined to each other of $$\bar{a}, \bar{b}$$ and $$\bar{c}$$ at an angle of $$60^o$$. Then $$|\bar{a}+\bar{b}+\bar{c}+\bar{d}|^2=?$$

In a plane at a given point $$\vec {A}$$ exist is vertically upward and $$\vec {B}$$ at point in north direction then find the direction at $$\vec {A}\times \vec {B}$$

Let $$\hat{a}, \hat{b}$$ and $$\hat{c}$$ be three unit vectors such that $$\hat{a}=\hat{b}+(\hat{b}\times \hat{c})$$, then the possible value(s) of $${|\hat{a}+\hat{b}+\hat{c}|}^{2}$$ can be:

If $$D\vec A = \vec a,$$  $$A\vec B = \vec b$$  and $$C\vec B = k\vec a$$ where $$k < 0$$ and X, Y are the mid-points of DB & DC respectively, such that $$\left| {\vec a} \right| = 17\& \left| {X\vec Y} \right| = 4$$, then k equal to -

$$3\overline {OD}  + \overline {DA}  + \overline {DB}  + \overline {DC}  = $$

Forces $$3 \vec { OA } $$, $$5 \vec { OB } $$ act along OA and OB. If their resultant passes through C on AB, then :

The angles of a triangles whose two sides are represented by vectors $$\sqrt(\vec { a } \times \vec { b })$$ and $$\vec{b}-(\vec { a }  \vec { b })\vec{a}$$ are in the ratio