Vector Algebra ...

A unit vector $${\vec a}$$ in the plane of $$\vec  = 2\hat i + \hat j$$ and $$\vec c = \hat i - \hat j + \hat k$$ is such that angle between $${\vec a}$$ and $${\vec b}$$ is the same angle between $${\vec a}$$  and $${\vec d}$$   where $$\vec d = \hat j + 2\hat k$$

ABCD is a parallelogram. The position vectors of A and C are respectively, $$3\hat{i}+3\hat{j}+5\hat{k}$$ and $$\hat{i}-5\hat{j}-5\hat{k}$$. If M is the mid-point of the diagonal DB, then the magnitude of the projection of $$\vec{OM}$$ on $$\vec{OC}$$, where O is the origin is?

If the unit vectors $$\vec{e}_1 \, and \, \vec{e}_2$$ are inclined at an angle $$2 \theta \, and \, |\vec{e}_1 - \vec{e}_2| < 1$$, then for $$\theta \in [0, \pi] , \theta$$ may lie in the interval

Let $$a, b$$ and $$c$$ be three unit vectors such that a is perpendicular to the plane of $$b$$ and $$c$$. if the angle between $$b$$ and $$c$$ is $$\dfrac { \pi  }{ 3 }$$, then $$|a\times b-a\times c|^{2}$$ is equal to 

If $$[\vec{a}\vec{b}\vec{c}]=2$$, then find the value of $$[(\vec{a}+2\vec{b}-\vec{c}(\vec{a}-\vec{b})(\vec{a}-\vec{b}-\vec{c})]$$.

Let $$\vec{a}=-\hat{i}-\hat{k}, \vec{b}=-\hat{i}+\hat{j}$$ and $$\vec{c}=\hat{i}+2\hat{j}+3\hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r}\times \vec{b}=\vec{c}\times \vec{b}$$ and $$\vec{r}\cdot\vec{a}=0$$, then the value of $$\vec{r}\cdot \vec{b}$$ is?

A non-zero vectors $$\overrightarrow{a}$$ is such that its projections along the vectors $$\dfrac{\hat{i}+\hat{j}}{\sqrt{2}}$$ and $$\dfrac{-\hat{i}+\hat{j}}{\sqrt{2}}$$ and $$\hat{k}$$ are equal then unit vector along $$\overrightarrow{a}$$ is

Let $$O$$ be the  centre a regular hexagon $$ABCDEF$$ Then the magnitude of sum of the vectors $$\overline { OA }, \overline { OB} ,\overline { OC },\overline { OD },\overline { OE }, $$ and $$\overline { OF } $$ is

The length of the projection of the line segment joining points $$(5, -1, 4)$$ and $$(4, -1, 8)$$ on the plane $$x+y+z=7$$.

If $$a$$ and $$b$$ are unit vectors along $$OA, OB$$ and $$OC$$ bisects the angle $$AOB$$. The unit vector along $$OC$$ is