Vector Algebra ...

$$ABCD$$ is a quadrilateral and $$E$$ the point of intersection of the lines joining the middle points of opposite sides. If $$O$$ is any point, then the resultant of $$OA,OB,OC$$ and $$OD$$ is equal to

The components of vector $$ i + j + k $$ along vector $$i+2j+3k $$ is

OAB is a given triangle such that $$\displaystyle \overrightarrow{OA}=\bar a$$, $$\displaystyle \overrightarrow{OB}=\bar b$$.
Also C is a point on AB such that $$\overrightarrow{AB}=2\overrightarrow{BC}$$. State which of the following statements are correct?

Let $$\displaystyle \overrightarrow{OA}=a$$ and  $$\displaystyle \overrightarrow{OB}=b$$ and $$\displaystyle \overrightarrow{OC}=a+b$$. What is the type of the quadrilateral $$OACB$$?

A vector $$a$$ can be written as

The projection of $$\bar a=3\hat i-\hat j +5\hat k$$ on $$\bar b=2\hat i+3\hat j+\hat k$$ is

The triangle $$ABC$$ is defined by the vertices $$A (1, -2, 2), B(1, 4, 0)$$ and $$C(-4, 1, 1)$$. Let $$M$$ be the foot of the altitude drawn from the vertices $$B$$ to side $$AC$$. Then $$\displaystyle \overrightarrow{BM}=$$

Given two vectors $$a=2\hat{i}-3\hat{j}+6\hat{k}$$, $$b=-2\hat{i}+2\hat{j}-1\hat{k}$$ and $$\displaystyle \lambda$$ = ratio of the projection of $$a$$ on $$b$$ and the projection of $$b$$ on $$a,$$ then the value of $$\displaystyle \lambda $$ is

Let $$a=i+2j+k, \ b=i-j+k$$ and $$c=i+j-k $$. A vector in the plane of $$ a$$ and $$b$$, whose projection on $$c$$ is $$\displaystyle \frac{1}{\sqrt{3}}$$ is

Four forces of magnitude P, 2P, 3P and 4P act along the four sides of a square ABCD in cyclic order. Use the vector method to find the resultant force.