Vector Algebra ...

If $$|\overrightarrow{C}|^2=60$$ and $$\overrightarrow{C} \times (\widehat{i}+2\widehat{j}+5\widehat{k})=\overrightarrow{0}$$, then a value of $$\overrightarrow{C}\cdot (-7 \widehat{i}+2\widehat{j}+3\widehat{k})$$ is :

Let $$ABC$$ be a triangle whose circumcentre is at P.  If the position vectors of $$A, B, C$$ and P are $$\vec {a}, \vec {b}, \vec {c}$$ and $$\dfrac {\vec {a} + \vec {b} + \vec {c}}{4}$$ respectively, then the position vector of the orthocentre of this triangle, is:

The vectors $$\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$$ and $$\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$$ are the sides of a triangle $$ABC$$, then the  length of the median through $$A$$ is:

If C is the mid point of AB and P is any point outside AB, then 

In a parallelogram ABCD, $$|\overrightarrow{AB}| = a, |\overrightarrow{AD}| = b$$ and $$|\overrightarrow{AC}| = c$$, then $$\overrightarrow{DB}.\overrightarrow{AB}$$ has the value

Let $$P,\ Q,\ R$$ and $$S$$ be the points on the plane with position vectors $$-2\hat{i}-\hat{j},\ 4\hat{i},\ 3\hat{i}+3\hat{j}$$ and $$-3\hat{i}+2\hat{j}$$ respectively. The quadrilateral $$PQRS$$ must be a

Let  $$\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$$  and  $$\vec{c}=\hat{i}-\hat{j}-\hat{k}$$  be three vectors. A vector  $$\vec{v}$$  in the plane of   $$\vec{a}$$ and $$\vec{b}$$ , whose projection on  $$\vec{c}$$  is $$\displaystyle \dfrac{1}{\sqrt{3}}$$ , is given by $$;$$

The triangle $$ABC$$ is defined by the vertices $$A= (0,7,10)$$ , $$B=(-1,6,6)$$ and $$C=(-4,9,6)$$. Let $$D$$ be the foot of the attitude from $$B$$ to the side $$AC$$ then $$BD$$ is

The point $$C=(\dfrac{12}{5}, \dfrac{-1}{5},\dfrac{4}{5})$$ divides the line segment $$AB$$ in the ratio $$3:2$$. If $$B=(2,-1,2)$$ then $$A$$ is

$$ABCD$$ is a parallelogram and $$AC, BD$$ be its diagonals Then $$ \vec{AC} +\vec{BD}$$ is