Vector Algebra ...

If $$G$$ is the centroid of a  $$\Delta ABC$$, then $$\vec{GA} + \vec{GB} + \vec{GC}$$ is equal to

In triangle ABC, which of the following is not true.

Six vectors, $$\vec a$$ to $$\vec f$$ , all of magnitude 1 and directions indicated in the figure ( Consider all of them to be originating at origin ). Which of the following statement is true?

In triangle $$ABC$$, $$\angle A = 30^o$$, $$H$$ is the orthocentre and $$D$$ is the midpoint of $$BC$$. Segment $$HD$$  is produced to $$T$$  such that $$HD = DT$$. The length $$AT$$ is equal to

$$'I'$$ is the incentre of triangle of $$ABC$$ whose corresponding sides are $$a, b, c,$$ respectively, $$a\vec{IB} + b\vec{IB} + c\vec{IC}$$ is always equal to

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be unit vectors such that $$\vec{a} + \vec{b} - \vec{c} = 0$$. If the area of triangle formed by vectors $$\vec{a}$$ and $$\vec{b}$$ is $$A$$, then what is the value of $$4A^2$$?

In a trapezium, vector $$\vec{BC} = \alpha \vec{AD}$$. Also, $$\vec p = \vec{AC} + \vec{BD}$$ is collinear with $$\vec{AD}$$ and $$\vec p = \mu \vec{AD}$$, then which of the following is true?

$$P(\vec{p})$$ and $$Q(\vec{q})$$ are the position vectors of two fixed points and $$R(\vec{r})$$ is the position vector of a variable point. If $$R$$ moves such that $$(\vec{r} - \vec{p}) \times (\vec{r} - \vec{q}) = \vec{0}$$, then the locus of $$R$$ is

$$A, B, C$$ and $$D$$ have position vectors $$\vec{a}, \vec{b}, \vec{c}$$ and $$\vec{d}$$, respectively, such that $$\vec{a} - \vec{b} = 2 (\vec{d} - \vec{c})$$, then

Let $$ABC$$ be a triangle whose centroid is $$G$$, orthocentre is $$H$$ and circumcentre is the origin '$$O$$'. If $$D$$ is any point in the plane of the triangle such that no three of $$O, A, C$$ and $$D$$ are collinear satisfying the relation $$\vec{AD} + \vec{BD} + \vec{CH} + 3 \vec{HG} = \lambda \vec{HD}$$, then what is the value of the scalar $$'\lambda'$$?