If one side of a square is represented by the vector $3i + 4j + 5k,$ then the area of the square is

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$,then $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$ does not exceed:

Let the vectors $\overrightarrow{AB} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\overrightarrow{AC} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ be two sides of a triangle $ABC$. If $G$ is the centroid of $\triangle ABC$, then $\frac{27}{7}(\overrightarrow{AG})^2 + 5 =$

Let $\vec{a}$ and $\vec{b}$ be non-collinear vectors. If the vectors $(\lambda-1) \vec{a}+2 \vec{b}$ and $3 \vec{a}+\lambda \vec{b}$ are collinear, then the set of all possible values of $\lambda$ is

The vector in the direction of vector $5 \hat{i} - \hat{j} + 2 \hat{k}$ with a magnitude of $8$ units is:

If $\vec{a} = \hat{i}+\hat{j}-\hat{k}$ and $\vec{b} = 2\hat{i}-3\hat{j}+\hat{k}$ are adjacent sides of a parallelogram,then the lengths of its diagonals are