If $a, b, c$ are non-zero vectors such that $a \cdot b = a \cdot c$,then which statement is true?

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

If the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is $1,$ then $\lambda$ is equal to ..... .

If $\vec{a}$ is a unit vector and $(\vec{x}-\vec{a}) \cdot (\vec{x}+\vec{a}) = 8$,then $|\vec{x}| = $ . . . . . . .

Let $\vec{\alpha}=\hat{i}+\hat{j}+\hat{k}$, $\vec{\beta}=\hat{i}-\hat{j}-\hat{k}$ and $\vec{\gamma}=-\hat{i}+\hat{j}-\hat{k}$ be three vectors. $A$ vector $\vec{\delta}$, in the plane of $\vec{\alpha}$ and $\vec{\beta}$, whose projection on $\vec{\gamma}$ is $\frac{1}{\sqrt{3}}$, is given by

In a $\triangle ABC$,let $G$ denote its centroid and let $M, N$ be points in the interiors of the segments $AB, AC$,respectively,such that $M, G, N$ are collinear. If $r$ denotes the ratio of the area of $\triangle AMN$ to the area of $\triangle ABC$,then